Eindhoven University of Technology

MASTER

Direct lift control for the Cessna Citation II

Gerrits, M.

Award date:1995

Link to publication

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

DEPARTMENT OF ELECTRICAL ENGINEERING

Measurement and Control Group

Direct Lift Controlfor the

Cessna Citation II

by M. Gerrits

tl8Eindhoven University of TechnologyMeasurement and Control GroupDen Dolech 25600 MB EindhovenThe Netherlands

National Aerospace Laboratory NLRInformatics divisionAnthony Fokkerweg 21059 CM AmsterdamThe Netherlands

(TUE/ER)(TUE/ER)(NLR/IW)(NLR/VM)

M.Sc.Thesis carried out at the National Aerospace Laboratory (NLR)from April 1, 1994 to September 30, 1994

commissioned by prof. P.P.J. van den Boschunder supervision of: dr. S. Weiland

drs. A.A. ten Damir. J.C. Terlouw

date: 11-14-1994

The Department of Electrical Engineering of the Eindhoven University of Technology accepts

no responsibility for the contents of M.Sc.Theses or reports on practical training periods.

- 1 -

~ Abstract~ ----------------------------------~

Abstract

Direct lift control might improve passenger comfort in aircraft flying in turbulent air.This task is nowadays performed by using conventional control surfaces, i.e. elevator,aileron and rudder. The use of these control surfaces is termed moment control tech­nique. A change in flight path is made by changing the moment equilibrium, whichindirectly produces forces for controlling the motion of the aircraft. Thus, there is acoupling between aircraft flight path and attitude. Consequently, there are inherentlimitations as far as separate control of attitude and flight path is concerned. Directlift control provides a direct means for producing forces thereby influencing the momentequilibrium slightly. Such a capability removes the limitations caused by the coupling ofattitude and flight path control, and offers novel and unique modes of aircraft control,such as flight path control with constant angle of attack, decoupled flight path/ altitudecontrol and decoupled pitch/altitude control. To enable these control modes, specialcontrol surfaces are needed, which can change lift and drag in both upward and down­ward direction. For high frequency control they must be able to operate very quickly.Undesirable pitching moments introduced by the direct lift control surfaces have to becompensated by appropriate elevator deflection angles.

The additional control modes offer benefits for gust alleviation. The most unpleasantmotions as experienced by passengers are vertical motions and vertical accelerations.Variations in forward speed or lateral motions are also unpleasant, but these motionscannot be controlled using direct lift surfaces. The symmetric motions of flight areconsidered, since only vertical gusts can be alleviated by direct lift control. The con­tributions of this study to the research project are the following: The nonlinear modelof the Cessna Citation II light jet aircraft has been modified. Nonlinear models of thespeedbrakes and the flaps are included. Conventional flaps and speedbrakes are used asdirect lift surfaces. Only slight modifications to the actuators are proposed, concerninga linear deflection range and enlarged deflection rates for high frequency operation.

The performance of a proportional feedback direct lift controller for gust alleviation hasbeen evaluated, and the results look promising. Improvements in vertical acceleration bya factor 2 are achieved using combined elevator/flaps control. However, this is attendedby an increase in drag, resulting in variations in horizontal speed. The results mayfurther be improved using special direct lift control flaps at the trailing edge of theconventional flaps, capable of moving much faster than conventional flaps.

- III -

M.Sc. thesis subjectIf""

@------------~

Automatic flight control during flight in turbulent air

Active Flight Controls/National Fly-by-wire Testbed project

AFC2

M.Sc.

Subject title:

Project:

Subproject:

Mentors:

Professor:

Location:

Goal:

Start date:

End date:

thesis subject

A.A. ten DamJ.C. TerlouwS. Weiland

P.P.J. van den Bosch

National Aerospace Laboratory (NLR)Department of Mathematical Models andMethods, Informatics Division

Design of an automatic flight control lawfor the Cessna Citation II for gust alleviation

April 1, 1994

September 30, 1994

(NLR/IW)(NLR/VM)(TUE/ER)

(TUE/ER)

Description

Recently, the National Aerospace Laboratory (NLR) and the Delft University of Tech­nology (TUD) have started the Active Flight Controls/National Fly-by-wire (FBW)Testbed project (AFC/NFT). The main goal of this research project is to gain a know­ledge base in the area of FBW control technology, large enough to allow a future ge­neration of Dutch-built aircraft to make use of FBW techniques. For this and otherpurposes, NLR and TUD jointly operate a Cessna Citation II (the successor of theCessna Citation 500) light jet aircraft, which will be configured as NFT.

When an automatic control system is applied for improving passenger comfort withinaircraft flying in turbulent air, it is common practice that this system controls thevertical speed and the pitch angle. From previous investigations it is concluded thatdue to a non-minimum phase response of height to an elevator deflection, the elevator isless appropriate for gust alleviation. A possible solution to improve the height response

- IV -

~ Direct lift control for the Cessna Citation II~ ----------------------------------~

of the aircraft is to use the flaps and/or the speedbrakes as welL This technique isknown as direct lift control (DLC).

For the actuation of these control surfaces, continuous-time models have to be designed.Subsequently, a comparison of the possible strategies to improve passenger comfort,i.e. combined elevator/speedbrakes control or combined elevator/flaps control, must beperformed.

The work will be carried out at the NLR and the TUE and will consist of:

1. a critical review of the available control algorithms and literature on aircraft ridesmoothing systems and Direct Lift configured aircraft;

2. a classification of disturbances acting on the aircraft, which are relevant for pas­senger comfort;

3. a description of turbulence;

4. a specification of actuator models for the flaps and the speedbrakesj

5. design and comparison of combined elevator/speedbrakes control against combinedelevator/flaps control by means of simulation studies;

6. investigations of the influence of disturbances on control performance.

- v ­Contents~e --------------------------------

,)'

Contents

Abstract

M.Sc. thesis subject

List of figures

Abbreviations

1 Introduction

2 Introduction to direct lift control2.1 A definition of passenger comfort2.2 Description of direct lift control .

2.3 Design choices and specifications

3 Extensions of the non-linear model for direct lift control3.1 Extension of the aerodynamics model3.2 Trim model .....3.3 Linearization model .3.4 Simulation model ..

4 Linear dynamics models4.1 Citation state space model .4.2 Turbulence state space model . . . . .4.3 Actuator dynamics and washout filter.4.4 Open loop system characteristics

5 PID controller design5.1 Elevator control .5.2 Combined elevator/flap control .5.3 Combined elevator/speedbrake control5.4 Simulation results . . . . . . . . . . ..

1

HI

VH

1

5

779

12

15

1518

19

19

23

23293234

3737

4041

43

- VI -

~ Direct lift control for the Cessna Citation II~ ----------------------------------~

6 Conclusions & Recommendations

A Linearized aircraft model

B Data plots of stability derivatives

C Simulink modelsC.l Description of modifications

D Open loop system characteristics

E Closed loop system characteristics

F Results of linear simulations

G Matlab files

87 pages total

47

53

55

5757

63

67

79

87

The original report is available at the NLR memorandum IW-94-034 and containsconfidential information. The version present at the Eindhoven University of Tech­nology is censored. The figures and tables in Appendix A, B, C, D and G containingconfidential information are omitted. For a full version of this report, please contactthe National Aerospace Laboratory NLR, Amsterdam, The Netherlands.

- Vll -

~ List of figures<e> -------------------------------,Jt::.

List of Figures

2.12.2

3.13.2

4.14.2

5.15.2

B.1B.2B.3B.4B.5B.6

Definition of flight path angle b), pitch angle (()) and angle of attack (a)Controller structure . . . . . . . . . . . . . . . . . . . .

Coefficients of forces in horizontal and vertical direcionStandard plant . . . . . . . .

General model representationDryden spectral densities.

Handling qualities ..Controller structure .

(Confidential) cdsaf: Increment in drag coefficient due to full 8sb

(Confidential) clsam: Increment in lift due to full 8sb

(Confidential) cmesaf: Increment in Cm6 due to full 8e • • • •.b

(Confidential) cmqsaf: Increment in Cm6 due to pitch rate ...b

(Confidential) cmsam: Incr. in pitching moment due to full 8sb

(Confidential) kss: Speedbrake effectiveness factor

1013

1718

2730

3838

555555565656

C.1 (Confidential) Overview of Simulink model layers 58C.2 (Confidential) cit-trim: citation trim model. . . . 58C.3 (Confidential) cit-lin: citation linearization model 58C.4 (Confidential) cit-ctrl: citation simulation model. 59C.5 (Confidential) CITATION: Main aerodynamics model 59C.6 (Confidential) AFM: Calculation of the aerodynamics coefficients 59C.7 (Confidential) rud-rudtrim coupling . . . . . . . . . . . . . . . . . 59C.8 (Confidential) aeromod: aerodynamics model of gust and controls 59C.9 (Confidential) cdcal: calculation of the drag coefficient .. 59C.10 (Confidential) deal: calculation of the lift coefficient. . . . . 59C.lI (Confidential) cmcal: calculation of the moment coefficient . 60C.12 (Confidential) Speedbrakes contribution to pitching moment 60C.13 (Confidential) Turbulence generator. . . . . . . . . . . 60C.14 (Confidential) actuator dynamics . . . . . . . . . . . . 60C.15 (Confidential) Linear model of DLC configured aircraft 60C.16 (Confidential) Elevator controller 60C.17 (Confidential) Flaps controller. . . . . . . . . . . . . . 61

- Vlll -

~ Direct lift control for the Cessna Citation II@y----------.)t:::.

D.1 (Confidential) System poles, short period mode .D.2 (Confidential) System poles, phugoid mode .D.3 (Confidential) Actuator with washout filter, step response.DA (Confidential) Dryden spectra: [WI, W3] to [ug,O:g]D.5 (Confidential) Bode plot ug to iL, hand q .D.6 (Confidential) Bode plot O:g to iL, hand q .D.7 (Confidential) Bode plot 8e to V and () .D.8 (Confidential) Bode plot 8e to 0: and q .D.9 (Confidential) Bode plot 8e to iL, hand qD.10 (Confidential) Bode plot 81 to V and () .D.ll (Confidential) Bode plot 81 to 0: and q .D.12 (Confidential) Bode plot 81 to iL, hand qD.13 (Confidential) Bode plot 8sb to V and () .D.14 (Confidential) Bode plot 8sb to 0: and q .D.15 (Confidential) Bode plot 8sb to iL, hand q.

E.1 Root locus elevator to pitch angle . . . . .E.2 Root locus elevator to pitch angle, phugoid modeE.3 Root locus elevator to pitch rate .EA Root locus elevator to pitch rate, phugoid mode .E.5 System poles, short period mode, inner loop activeE.6 System poles, phugoid mode, inner loop activeE. 7 Root locus flaps to vertical speed . . . .E.8 Root locus flaps to vertical acceleration .E.9 Root locus flaps to pitch acceleration .E.10 Root locus speedbrakes to vertical acceleration.E.ll Root locus speedbrakes to pitch accelerationE.12 Bode plot 8e to () and q ..E.13 Bode plot 81 to ~, h an~ q .E.14 Bode plot 8sb to hand h .E.15 Impulse response elevator to pitch angleE.16 Impulse response elevator to pitch rate .E.17 Impulse response flaps to vertical speed .E.18 Impulse response flaps to vertical accelerationE.19 Impulse response speedbrakes to vertical speed.E.20 Impulse response speedbrakes to pitch acceleration

F.1 Response of y,~, (), q to gust (inner loop active)F.2 Response of h, h", q to gust (inner loop active)F.3 Control activity (inner loop active) .FA Response of V, 0:, (), q to gust (DLC loop active)F.5 Response of iL, h", q to gust (DLC loop active)F.6 Control activity (DLC loop active) .

636364646464646464646565656565

6868696970707171727373747475757676777778

808081818282

- 1 -~ Abbreviations~--------------------------------~

Abbreviations

Scalars

A,B,C,D

aZcg

CDCL

Cmu' Cmw ' Cmq , .CXu ' CXw ' CXq , .CZu ' CZw' CZq , .c

Dcg

hJRDf{y

Lg

m

pqrSTtUo

U

Ug

U

[m]

[m][kg]

[rad 8-1]

[rad 8-1]

[rad 8-1 ]

[m 2]

[8][m 8-1 ]

[m8-1]

[m8-1]

State space matricesAcceleration in downward direction (Zb)Normal acceleration at the centre of gravityTotal non-dimensional drag coefficientTotal non-dimensional lift coefficientNon-dimensional stability derivativesNon-dimensional stability derivativesNon-dimensional stability derivativesMean aerodynamic chordNon-dimensional differential operator = VftGravitational constantHeight at centre of gravityRide discomfort indexNon-dim. radius of inertia about Y-axisGust wavelengthAircarft massAbbrev. of symmetric stability derivativesAngular velocity about the longitudinal axisAngular velocity about the lateral axisAngular velocity about the vertical axisWing areaTransformation matrixTimeHorizontal airspeedForward velocityHorizontal gust velocityNon-dimensional velocity in forward direction

- 2 -Direct lift control for the Cessna Citation II,./e --------------------

~

vwW

WI,W3

X,Y,Z = Xb,Yb,ZbXe,~,Ze

Xs,~,Zs

Xu,Xa,X(), ...XA

X cg

Zu, Za, Z(), ...

Vectors

[m][m]

True airspeedAircraft's weightDownward velocityWhite noise sources for gust simulationForward, side and downward body axesEarth coordinate system (north, east, down)Stability coordinate systemAbbrev. of symmetric stability derivativesPoint of measureLocation of the centre of gravity relative to cAbbrev. of symmetric stability derivatives

Input vectorState vectorOutput vectorDisturbance vector (known)

Greek

a [rad] Angle of attacka g [rad] Gust induced angle of attack

f3 [rad] Sideslip angle

I [rad] Flight path angleDa [rad] Aileron deflection angleDe [rad] Elevator deflection angleD1 [rad] Flap deflection angleDr [rad] Rudder deflection angleDsb [rad] Speedbrake deflection angle

/-lc Relative air density for symmetric motionp [kg m-3 ] Air density

aUg' a Wg Standard deviation of gust in horizontaland vertical direction

() [rad] Pitch angleTa [rad] Actuator time constantTn [rad] Washout :filter numerator time constantTw [sec] Washout :filter time constant

- 3 -..<.. AbbreviationsG----------~

Subscripts

acgDDLefgLm

ossbtw

Abbreviations

ActuatorCentre of GravityDragDirect Lift surfaceElevatorFlapsGustLiftMomentEquilibrium stateStability axis systemSpeedbrakeTrim surfaceWashout

AFCDDCDLCDSFCFBWMCTNFTNLRNLR/IW

NLR/VMRDIRMSTUDTUETUE/ERu.c.

Automatic Flight ControlsDirect Drag ControlDirect Lift ControlDirect Side Force ControlFly-By-WireMoment Control TechniqueNational Fly-by-wire Testbed projectNational Aerospace LaboratoryNLR, Informatics Division, Department of

Mathematical Models and MethodsNLR, Flight DivisionRide Discomfort IndexRoot Mean SquaredDelft University of TechnologyEindhoven University of TechnologyTUE, Measurement and Control sectionUnder Carriage

Matlab variables

CDSAF Increment in drag coefficient due to a full speed­brake deflection as a function of 0: (Figure B.l).

- 4 -~ Direct lift control for the Cessna Citation IIe----------~

CDSB

CLSAM

CLSBCMESAF

CMQSAF

CMSAM

CMSB

KSS

Cmqsb

Increment in drag coefficient due to a speedbrakedeflection.Increment in lift coefficient due to a full speedbrakedeflection as a function of 0: (Figure B.2).Increment in lift due to a speedbrake deflection.Increment in Cm6 due to an elevator deflection

sb

(Figure B.3).Increment in Cm6 due to a pitch rate (Figure

sb

B.4).Increment in pitching moment coefficient due to afull speedbrake deflection as a function of 0: (Fig­ure B.5).Increment in pitching moment coefficient due to aspeedbrake deflection.Speedbrake effectiveness factor as a function of thespeedbrake deflection angle (Figure B.6).

- 5 -~ Introduction~~ ----------------------------------~

1 Introduction

In this report, research is done on new technologies for gust alleviation using a modelof the Cessna Citation II. A solution to the gust alleviation problem is to use the flapsand speedbrakes as control surfaces, instead of using them as landing flaps and groundspeedbrakes only. This technique is known as direct lift control (DLC) and is usedin military aircraft for careful manoeuvring and positioning operations. Civil aircraftare rarely configured with direct lift control, since the requirements, indicated above,are less important for commercial aircraft. However, it can be used in civil aircraft toimprove ride smoothing.

The National Aerospace Laboratory (NLR) and the Delft University of Technology(TUD) have started the Active Flight Controls/National Fly-By-Wire (FBW) Testbedproject (AFC/NFT). The main goal of this project is to gain a knowledge base in thearea of FBW control technology, large enough to allow a future generation of Dutch­built aircraft to make use of FBW techniques. FBW means that there is no longera mechanical link between the pilot's control column and the control surfaces. Pilotcommands are translated into electrical signals, which are fed to actuators, moving thecontrol surfaces. For research on FBW, the TUD and NLR jointly operate a CessnaCitation II light jet aircraft, configured as NFT.

This report deals with the problem of designing a control system on simulation levelincorporating direct lift control (DLC) for improvement of the ride performance of theCessna Citation II light jet aircraft. Active control will be used to activate the controlsurfaces of the aircraft. Active control is defined here as the use of feedback controlto change the dynamic characteristics of the aircraft. There are basically two types ofbenefit, one associated with pilot handling characteristics, including tracking, 'carefree'manoeuvring, etc. and one associated with fully automatic control systems, independentof pilot input, including ride control, gust and load alleviation, etc. Active control alsoimplies the use of any available actuator, not only for the conventional control surfaces,but also for the direct lift and direct drag surfaces.

The main goal of this MS thesis subject is to investigate the applicability of directlift control on the Cessna Citation II light jet aircraft. Starting with the present-dayconfiguration of the controller system of the Cessna Citation II, a combination of controlsurfaces, giving the best results in minimizing a ride discomfort index, should be found.

- 6 -~ Direct lift control for the Cessna Citation IIe----------~

Several combinations will be analysed by linear model simulations. For this, nonlinearmodels of the Cessna Citation II are extended with the dynamics of the DLC-surfaces.A basis for this research project will be an analytical approach to the problem. Itis assumed that, with some modifications, the conventional flaps and speedbrakes canbe used as DLC surfaces. Structural modes of the aircraft are not considered in thesemodels. To gain insight into some of the problem areas, the design was first treated usinga simple model of the aircraft dynamics with additional first-order filters representing theactuation systems, including both elevator and DLC. Linear control laws are applied,with some small modifications, to a more thorough model of the aircraft, includingrepresentation of actuators, power controls, output equations and various nonlinearities.

Standard flaps and speedbrakes are used as direct lift surfaces. Only slight modificationsto the actuators are proposed, concerning a bounded deflection range and an enlargeddeflection rate for high frequency operation, which are minimum requirements for aDLC system. Another important requirement for the implementation of DLC is thatthe DLC surfaces may never cause the aircraft to go unstable. So, natural stability mustbe maintained.

This report is organized as follows. In chapter 2, a brief introduction to the principlesand newly available control modes of direct lift control will be presented. Some re­quirements for the control surfaces and design choices concerning the direct lift controlconfiguration will be given. A definition of passenger comfort is given, which is used asa criterion for controller performance. This criterion is a measure of ride discomfort andis called the Ride Discomfort Index (RDI). In chapter 3, a complete description of themodifications to the nonlinear model of the Cessna Citation II will be presented andthe underlying equations for lift, drag and moment are given and converted to a Matlabmodel. The difference between the simulation model and the linearization model will beexplained. The linear model is derived from the nonlinear model in chapter 4. The statespace model for symmetric motions and the state space model for turbulence are de­rived. First order models of the various actuators are presented. The open loop systemcharacteristics, i.e. pole/zero map, frequency response and phase response are presentedin section 4.4. In chapter 5, four different controllers are designed: elevator control only,combined elevator/flaps control, combined elevator/speedbrakes control and combinedelevator/flaps/speedbrakes control. The results of the simulations are presented anddiscussed. Finally, in chapter 6, conclusions are drawn with respect to the effectivenessof the DLC-system for gust alleviation and the applicability of this system in the CessnaCitation II.

- 7 -~ Introduction to direct lift control~--------,jt::.

2 Introduction to direct lift control

In this chapter, the control technique Direct Lift Control (DLC) will be dis­cussed with some new flight modes, which come available with the applica­tion of these techniques. Before a controller for gust alleviation incorporatingDLC can be designed, some criterion of passenger comfort, or discomfort,should be defined. Two criteria are given in section 2.1. The newly avail­able flight modes are discussed in section 2.2. Some design choices, to bemade when DLC for gust alleviation is implemented in the Cessna CitationII model, are described in section 2.3.

2.1 A definition of passenger comfort

To design a controller, which improves the ride comfort of passengers inside an aircraft,one has to know which levels of disturbance are acceptable and which levels are not.Vertical accelerations are mostly unpleasant for passengers and therefore the verticalacceleration needs to be reduced. As a measure of discomfort, a Ride Discomfort Index(RDI) is frequently used. Several definitions of the RDI are given in the literature,e.g. [McLean'90] and [Erkelens'74]. In both cases, the vertical acceleration is a criticalfactor. Ride improvement is defined as the reduction of the normal acceleration responseto vertical turbulence, measured at specified positions along the fuselage.

Considering only the symmetric motions, the RDI as derived in [Erkelens'74] can berewritten as:

JRD = 1.8 + 17.5az + 2.45q (2.1)

where az and qare the root mean squared (RMS) value of the vertical acceleration atthe centre of gravity and the RMS value of the pitch acceleration respectively. Equation(2.1) is the approximation of a frequency dependent ride discomfort function. The RMSvalue of a signal is defined as:

(2.2)

Some levels of comfort, based on passenger experience, are given by:

- 8 -Direct lift control for the Cessna Citation II~

~ ---------------------

1 = very comfortable2 = comfortable3 = neutral4 = uncomfortable5 = very uncomfortable

In [McLean'90] another RDI is given by:

(2.3)

where CLa = 8CL /8n and k a constant of proportionality and W/ S the wing load factor.CL is the total wing lift coefficient. JRD is a scalar that represents the ratio of wing liftslope to wing loading. The stability derivative Zw, involved with CLa , is a measure forchange in normal force due to changes in heave velocity [McLean'90]:

-pSV -pUogZw = 2m (CLa +CD) ~ 2k JRD (2.4)

in which CD is the total drag coefficient, p the air density, S the wing area, m theaircraft mass, V the true airspeed and Uo the speed in forward direction.

If it is assumed that CD ~ CLa and Uo ~ V, then the approximation in (2.4) isvalid. For the short period motion, the differential equation in vertical speed w holds:tV = Zww + Uoq + L ZOi8j, where q is the pitch rate, Zj = ~r the stability derivativeof the ith control surface 8j , i being {elevator, flap, speedbrake}. An expression for thevertical acceleration at the centre of gravity (a ZCg ) is given by:

aZcg tV - Uoq = Zww +L ZoA

-i~OgJRDw+ L Zo;8j(2.5)

It may be clear that minimizing aZcg and the control surface effort is equal to minimizingJRD , since all other terms in (2.5) are constant. The RDI is directly dependent onvertical acceleration. After a change in lift, the initial vertical acceleration h(h ~ -az )

is proportional to the difference between lift L and weight W:

(!h) = !:- _1 = 6.L9 t=to W W

(2.6)

where L = p~2S CL and to the time at which the lift to weight ratio started to change.With constant angle of attack, the initial vertical acceleration is proportional to thedirect lift device deflection angle:

(1 .. ) pV2S-h "-' 2W CL6DL 8DL + influence of other surfaces9 t=to

(2.7)

- 9 -~ Introduction to direct lift control<@y------------~

So, DLC should be a proper technique for vertical gust alleviation, since DLC directlycontrols the lift by that controlling the vertical acceleration.

The vertical acceleration at a location x A can be calculated from the states and thestate derivatives:

aZA = Uo(6: - q) - xAq = aZcg - xAq (2.8)

where XA is the location in the aircraft at distance XA in front of the centre of gravity.The vertical acceleration at location XA (azJ is an output scalar and will be added to theequations of symmetric motion. It is chosen to calculate the RDI with the RMS-valuesof the variables, so all frequency components are weighted equally.

Approximated values for XA:

pilot seat +8 mfront seat +5 mcentre of gravity 0 maft seat -5 m

2.2 Description of direct lift control

Aircraft control with the use of conventional control systems, such as elevators orailerons, can be considered a moment control technique (MCT). Via a change of themoment equilibrium, MCT indirectly produces forces for controlling the motion of theaircraft. Furthermore, MCT yields a coupling of attitude and flight path control. Con­sequently, there are inherent limitations as far as separate control of attitude and flightpath is concerned. The control technique termed 'Direct Lift Control' (DLC) providesa direct means for producing forces with small influence on the moment equilibrium.Such a capability removes the limitations caused by the coupling of attitude and flightpath control, and offers novel and unique modes of aircraft motion.

Translational motion of the aircraft is characterized by vertical translation, lateral trans­lation and longitudinal translation. A possibility for controlling these translational mo­tions is to use DLC surfaces to produce lift forces, direct side force control (DSFC)surfaces to produce side forces and direct drag control (DDC) surfaces to produce dragforces.

Conventional flight path control with the elevator yields two effects that may be adverse.The first consists of the delay between the initiation of the control action and flight pathresponse. As said before, this is because a change in moment equilibrium is necessary forproducing a force. The second adverse effect is that the force produced at the tail due toan elevator deflection is opposite to the flight path change command. Consequently, theaircraft initially moves in the wrong direction. Furthermore, rotational pitch dynamicscause a delay in the change of lift. This in turn results in a delay in altitude change,which is even more slowed down due to the two integration steps between lift and altitudechange, i.e. h = V sin / :::::: V, = ~ f !:i.L dt and !:i.h = f h dt, where / is the flight pathangle. For / ~ 1 ---7 sin / :::::: /.

- 10 -~ Direct lift control for the Cessna Citation II~ --------------------------------~

With DLC, some additional flight control modes become available. The flight path angle(,), the pitch angle (B) and the angle of attack (a) are defined according Figure 2.1:

Figure 2.1: Definition of flight path angle (,), pitch angle (B) and angle of attack (a)

X b is the body axes system, X s the stability axes system, X e the earthreference system and V the true airspeed.

The new modes are:

1. Flight path control with constant angle of attack.Flight path angle and pitch attitude are changed, with angle of attack remainingconstant: a = B- , =const. Thus, this mode can be characterized by:

(2.9a)

(2.9b)

and the direct lift command is used for changing flight path. The elevator is usedfor a change in attitude. In this mode the attitude of the body axes system andthe stability axes system relative to the earth reference system is changed.

2. Decoupled flight path/attitude control.In this mode, the aircraft can change its flight path angle without a change inattitude: B = , + a =const. Hence, there follows:

(2.10a)

(2.10b)

In this case, the direct lift command must exceed the opposite lift change dueto ~a so that a net lift change remains for flight path control. In this modethe attitude of the stability axes system relative to the earth reference system ischanged, with the body axes system remaining fixed.

3. Decoupled pitch/altitude control.In this mode, also called 'fuselage pointing', altitude can be controlled without

- 11 -~ Introduction to direct lift control@----------.J'

changing flight path, i.e. I = () - a =const. Leading to

o!.l()

(2.11a)

(2.11b)

The direct lift compensates for a lift change due to !.la, induced by DLC surfacedeflection. This mode can also be characterized by:

(2.12a)

(2.12b)

In this mode the attitude of the body axes system relative to the earth referencesystem is changed, with the stability axes system remaining fixed.

The latter mode will be considered in this report. For DLC, special control surfaces,having the following properties are needed:

1. Changes in lift (and drag) in positive and negative direction;

2. Quick operation for high frequency control;

3. Only small undesirable pitching moments.

The pitching moments have to be compensated by proper elevator deflection. Thesemoments should be small, because the elevators should be used primarily for manoeu­vring commands. For manoeuvring, it is recommendable to have the disposal of the fulldeflection range of the elevators.

For vertical flight path control, DLC provides the capability of producing lift instan­taneously. This cannot be achieved by conventional elevator control, since it shows adelay in lift buildup which results in an even greater delay of flight path response. Anincrease in lift, caused by atmospheric disturbance can be eliminated by a proper de­flection of the direct lift device, such that the resulting lift remains constant (!.lL = 0).The DLC-surfaces will be operated about a neutral point, so they can follow positiveand negative deflection commands.

In the following chapters it will be shown that by using DLC surfaces in the Citationmodel, separated pitch/altitude control can almost be achieved, which is advantageousfor gust alleviation with DLC. It will also be shown that the DLC surfaces produce onlysmall pitching moments, in comparison with the pitching moments introduced by theelevators.

Direct drag control (DDC) provides the capability of producing drag instantaneously.An increase in lift, caused by atmospheric disturbances can be alleviated by a properdeflection of the direct drag device. The increased drag will reduce the lift to drag ratio,causing the aircraft to accelerate downwards. The DDC-surfaces will also be operatedabout a neutral point.

- 12 -Direct lift control for the Cessna Citation II,/e ----------------------------------

~

2.3 Design choices and specifications

There are many configurations possible in designing a direct lift control system. In thisreport, a particular configuration was chosen. Because this research covers the firststeps of a feasibility study about the implementation of direct lift control in the CessnaCitation II, a simple configuration, based on control surfaces already present on theaircraft, will be used. If these surfaces could be used, no costly modifications to theaircraft itself are needed. Only the actuation system, driving the DLC surfaces shouldbe adapted.

Appropriate surfaces for direct lift control are the flaps. Extending the flaps will in­crease the wing lift. Appropriate surfaces for direct drag control are the speedbrakes.Extending the speedbrakes will increase the drag. The extra control surfaces will havetheir influence on the natural stability of the aircraft. The speedbrakes give an increasein phugoid damping, since the drag increases. Thus, the speedbrakes have a positiveeffect on the natural stability. When the flaps are used, the lift will increase, thereby de­creasing phugoid damping. Using these surfaces, especially the speedbrakes, introducesdrag in horizontal direction. So, another option is to use the speedbrakes for horizontalspeed control, but no attention will be paid to this subject here.

Only the symmetric motion will be considered, because DLCjDDC is mostly related tovertical accelerations. The controller is assumed to make use of three control surfaces:the elevators, flaps and speedbrakes. No difference is made between the left and rightwing flaps, speedbrakes and elevators. So, there will only be one control signal percontrol surface pair. Activating the left and right control surfaces independently wouldalso generate asymmetric motions.

The structure of the controller to be designed will consist of three independent parts asdepicted in Figure 2.2:

1. An elevator loop, controlling the attitude by activating the elevators. The inputsto the elevator controller are the pitch angle and the pitch rate.

2. A flaps loop, controlling the altitude by activating the flaps. The inputs to theflap controller are the vertical speed, vertical acceleration and pitch acceleration.

3. A speedbrakes loop, controlling the altitude by activating the speedbrakes. Thiscontroller uses the same inputs as the flap controller.

A separate elevator loop for attitude control is a very common design approach inautomatic flight control systems (AFCS) design [Ruigrok'92]. For safety reasons, theelevator loop is designed as a stand alone controller, augmenting the aircraft stability.The other loops are separated for the analysis of combined elevator jflaps and combinedelevatorjspeedbrakes control, without interaction between the flaps and the speedbrakescontrollers. It is, however, very likely that much better results can be obtained usingan integrated controller for both flaps and speedbrakes, or even for all three surfaces.

- 13 -~ Introduction to direct lift controle> ----------------------------------,jt::.

gust

aircraftmodel

Kf

comfortcriteriaoutput

(8, q)

(h,h,q)

Figure 2.2: Controller structure

The controllers, used for gaining insight in DLC, consist of pure proportional feedback,because with this type of feedback, most initial effect is to be expected. Proportionalcontrol can show the effects of DLC well enough to see if DLC can and should be usedfor passenger comfort improvement.

When using the DLC surfaces, a small pitching moment will be introduced, which shouldbe compensated by the elevators. Therefore, there exists some interaction between thecontrollers of the flaps/speedbrakes and the controller of the elevator. Similarly, there isan interaction between the flaps and the speedbrakes. These controls each have the sameeffect on vertical speed, but in opposite directions, i.e. a positive flap deflection will besupported by a negative speedbrake deflection. A considerable interaction between bothflaps and speedbrakes and the elevator is the downwash lag effect, which is created by aflap and/or speedbrake deflection. This effect consists of a change in air-current createdby the DLC surfaces, which changes the angle of attack of the elevators. The time-delaybetween the wing and the horizontal stabilizer is not taken into account.

Separating the three controllers cannot be done without loss of generality. The smallpitching moment introduced by the direct lift control surfaces must, somehow, be cor­rected by proper elevator deflection. Therefore, a cross coupling between the flap con­troller and the elevator controller is included. It is assumed that this pitching momentis proportional to the DLC surface deflection. The pitching moment introduced by thespeedbrakes is corrected the same way.

If the problem is concerned with an elastic aircraft, in which structural bending is sig­nificant, the sensor dynamics can be significant and the full mathematical model shouldbe used. In the citation model most sensors are accelerometers or gyroscopes. Thelocation of these sensors on the fuselage affects the performance of the DLC controllermore strongly than a sensor's dynamics. It is, however, assumed that all sensors arelocated in the centre of gravity of the aircraft, having no time-delay and no noise. Thevertical acceleration at the pilot seat will be calculated from the signals generated bythe vertical accelerometer and pitch accelerometer, located in the centre of gravity. An­other important simplification is neglecting the structural bending modes of the aircraft,

- 14 -~ Direct lift control for the Cessna Citation II~ ----------------------------------~

while some of these modes are within the frequency range where gust appears. Somestructural modes appearing within the gust frequency range are [Erkelens'75]:

Structural mode Damped Frequency (Hz) Damping1. First wing bending (vertical) 2.7 0.12. First fuselage bending 5.0 0.033. First wing torsion 5.7 0.014

Neglecting these modes may cause extremely great forces on the aircraft structure.When the elevator is moved down and flaps are extended simultaneously, the fuselagetends to bend down between the wing and the horizontal stabilizer. In the ultimate casethe fuselage could break. When the flaps, located at the trailing edge of the wings, or thespeedbrakes, located above and aft of the centre of gravity, are extended, a rotationalforce about the aerodynamic centre is induced, causing a torsion of the wing. The extralift generated by the flaps may cause the wing to bend up at the wing tips. In theultimate case the wing could be ripped off the aircraft.

Control problem formulationFor this research the conventional flaps and speedbrakes are projected as DLC surfaces.To obtain the required deflection rates special DLC surfaces located on the trailingedge of the flaps should be used, since the conventional flaps cannot follow these rapidmotions. For now, it will be assumed that the conventional flaps and speedbrakes areused as DLC and DDC surfaces. However, the actuators implemented in the Citationnow are assumed to be replaced by actuators with a higher bandwidth.

The Cessna Citation II is theoretically configured for use of DLC surfaces. For that, thespeedbrakes and the flaps may take any deflection angle between zero and it's maximumdeflection. The maximum rate of these control surfaces is specified so that the DLCsurfaces can operate within the entire gust bandwidth. The structural modes and thetime-delay between the wing and the horizontal stabilizer will not be considered. Forthis configuration, a controller has to be designed that reduces the vertical accelerationat arbitrary points of measure on the fuselage. The flight path angle has to remain asconstant as possible. The relative damping of both modes of flight (phugoid and shortperiod) should be at least 0.5 for handling qualities.

- 15 -~ Extensions of the non-linear model for direct lift controle----------

.)tC.

3 Extensions of the non-linearmodel for direct lift control

In this chapter the nonlinear model of the Cessna Citation aircraft will beextended with the aerodynamics of the speedbrakes. In section 3.1 the aero­dynamics of the speedbrakes is added to the nonlinear model. This modelis implemented in Simulink, a Matlab toolbox (the reader is referred to[Simulink] and [Van der Linden'94]). The structure of this model is outlinedin Figure C.1 The input vector had to be extended, because the input vec­tor was not supplied with control signals for the flaps and the speedbrakes.This model contains the main aerodynamics equations of the Cessna Cita­tion. The models for trimming, linearization and simulation are high levelmodels, using the nonlinear aerodynamics model. In section 3.2, the trimmodel of the Cessna Citation will be described. The trim routines calculatethe correct surface deflections to keep the aircraft in a steady flight whenthe pilot does not move the control column. In section 3.3, the lineariza­tion model of the Cessna Citation will be described. With this model, thenonlinear model of the Citation can be transformed in a linear model, whichcan be used for controller design. In section 3.4, the simulation model willbe described. In this model, the DLC controller (to be designed in section5), actuator dynamics and output filters are present, forming a closed loopwith the aircraft.

Important The nonlinear model of the Cessna aircraft, offered by the Delft Universityof Technology, contains the aerodynamic data and stability derivatives of the CessnaCitation 500. The Cessna Citation 500 is the predecessor of the Cessna Citation II.Therefore, any change in aerodynamics characteristics as done in the following sectionsis based on the aerodynamics of the Cessna Citation 500 aircraft.

3.1 Extension of the aerodynamics model

The nonlinear model of the Cessna Citation aircraft, as it is now, is not configuredfor DLC. Therefore, the model representing the aerodynamics of the aircraft has to be

- 16 -...<. Direct lift control for the Cessna Citation II@J----------,J.:::.

adapted. The aerodynamics of the flaps is already present (implicit), so that will notbe considered now. The speedbrakes, however, are not included in the model. Thespeedbrakes and flaps introduce pitching moments, because they contribute to the winglift at distance (X C9 - x w ) from the centre of gravity, so they affect not only the verticalmotions, when symmetry is assumed. The DLC surfaces initiate the following effects:

1. A pitching moment, which creates a rotational motion about the lateral axis;

2. A translational force in vertical direction;

3. A translational force in horizontal direction.

For the aircraft, flying at sufficient high altitudes (no ground effect) and with the un­dercarriage retracted, the mathematical model used to build up the total aerodynamicforce and moment coefficients can be calculated by respectively [Broos '87]:

CLbasic + C Loe + CLoSb + C Lq + C Lo

CDbasic + C Doe +C DOSb + CDq

(3.1 )

where Cm is the pitching moment coefficient, CL the total lift coefficient and CD thetotal drag coefficient. The indices have the following meaning:

basic basic aircraft lift coefficientDe elevator deflection angle [rad]Dsb speedbrake deflection angle [rad]q pitch rate [rad s-l]

The coefficients consist of a basic aircraft part and some parts that contribute to theforce, due to a deflection of a control surface or due to motions that deviate from equi­librium flight. In (3.1), only the contributions of surfaces and motions in the symmetryplane are calculated. Therefore, there will be some model uncertainties due to not con­sidered contributions in the coefficients. It attracts attention that no explicit coefficientfor the flaps is present. The influence of the flaps on the moment, lift and drag coefficientis implicitly calculated in the contribution of the other surfaces, e.g. the characteristicsof the contribution in lift due to a full elevator deflection are altered when the flaps aremoved to another position.

The coefficients in forward and downward direction can be calculated from (3.1) (seeFigure 3.1):

Cx

Cz(3.2)

or:

(3.3)

- 17 -~ Extensions of the non-linear model for direct lift controle----------,)'

-CZD..---=----~~----r=---.Xb

-CXD

Figure 3.1: Coefficients of forces in horizontal and vertical direcion

The coefficient of the pitching moment remains the same. The increment in lift, dragand moment due to a speedbrake deflection will be given attention to here. For theother incremental coefficients the reader is referred to [Broos'87].

The increment in the drag coefficient due to a speedbrake deflection is given by:

(3.4)

where ]{sb(8sb ) is the speedbrake effectiveness factor as a function of the speedbrakedeflection angle 8sb (see Figure B.6). CDSb(a) is the increment in the drag coefficientdue to a full speedbrake deflection as a function of a (see Figure B.1 and [Broos'87]).Apparently, CD6 is not dependent on Mach! In steady conditions, the change in lift

sb

is not linear with the speedbrake deflection angle. At small speedbrake angles theairflow over the speedbrakes re-attaches to the aerofoil surface ahead of the trailingedge, with little change in total lift. Only when the separated flow region extendsbeyond the trailing edge, there is a significant loss in lift. In Figure B.6 the nonlinearityis approximated by two linear parts, one with a low slope in the small speedbrake angleregion and one with a high slope in the greater speedbrake angle region.

The increment in the lift coefficient due to a speedbrake deflection (8sb ) is given by:

(3.5)

where CLsb is the increment in lift due to a full speedbrake deflection as a function of a(see Figure B.2).

- 18 -Direct lift control for the Cessna Citation II.r'e ----------------------

~

The increment in the moment coefficient due to a speedbrake deflection is given by:

(3.6)

where Cm6eSb (a) is the increment in Cmsb due to an elevator deflection (be) (see Figure

B.3). Cmqsb(a) is the increment in Cmsb due to a pitch rate (q) (see Figure B.4). Cmsb(a)is the increment in the pitching moment coefficient due to a full speedbrake deflectionas a function of a (see Figure B.5).

The modifications to the aerodynamics model are described in Appendix C.l. Thismodel represents the standard plant which will be used in the trim model, the lineariza­tion model and the simulation model. The standard plant is presented in Figure 3.2.

State Xderivative .}~

State X

Flight path

aerodynamicsmodelof the

Citation

!! { Controls

EnginesI

Massinit '1 _

d Gust

Figure 3.2: Standard plant

3.2 Trim model

With the underlying systems being modified, the model is configured for Direct LiftControl. With the trim model (Figure C.2), the aircraft can be trimmed to a steadyflight with certain initial conditions. Trimmed means that the aircraft will fly accordingto a constant flight path, e.g. straight and level flight, which will be considered here.The uppermost layer of the trim model (Figure C.2), is modified in the way that thespeedbrakes are part of the control vector and the flap and speedbrake neutral angles areread from the Matlab workspace (variables flap and speedbrake). The input vector ofthe trim model is given by:

The Matlab routine callmg this model is called trim_ac .m. The output of this model isthe state vector. The Matlab script file for trimming the aircraft has also been modified.

- 19 -~ Extensions of the non-linear model for direct lift control<e> ----------------------------------~

It is possible to enter the initial flap and speedbrake deflection angle. Besides, the modeldatafile "data\ citdlc. mat" is read immediately, instead of prompting for each filename.

The outputs of the trim routine are the initial flight conditions which make the aircraftfly according to a pre-specified path, when no control signals are provided to the controlvector. The results are stored in a Matlab binary file with extension .TRI. This file isneeded for linearization and simulation. In Appendix G it is explained how to start thetrim routine.

3.3 Linearization model

For controller design, a linear representation of the aircraft model is used. With thelinearization model (Figure C.3), the nonlinear model is prepared to be linearized. Thelinearization model differs at a few points from the trim model. The input vector hasbeen changed from five to nine inputs, which are

(3.7)

in which all inputs, needed for the symmetric model, are present, as will be seen in thenext chapter. The state vector consists of twelve states in which all states related tothe symmetric motions ([V,a,B,q]) are present:

K = [p q r V a f3 ¢ B 7/J h x y]T (3.8)

The output vector has been extended so that all measurements, which are needed bythe controller, are present. The output vector is:

11 = [p qr V a f3 ¢ B 7/J h x y hh q] T (3.9)

The Matlab routine calling this model is called lin_ac. m. This routine has been changed,so it can deal with the extra inputs needed for DLC. If a Matlab variable trim exists,the trim file, containing the working point of the model, will be loaded from . \ dat a \and the linearization is started immediately. With this option, it is possible to linearizemultiple conditions at once. The variable trim must contain the name of the trim file,generated by the Matlab routine trim_ac without extension. The output of this routineis a linear state space model representation of the aircraft. This model is written toa Matlab binary file with extension . LIN, which can be used for controller design andlinear simulations.

3.4 Simulation model

The simulation model citctrl (Figure CA) contains the nonlinear model of the Cita­tion, the output equations, filters, actuators and a controller. The equations for the

- 20 -~ Direct lift control for the Cessna Citation II~ ----------------------------------~

sensors are obtained from the Citation 500 model from [Bogers'93]. The outputs are thesame as described in section 3.3. In the block trim conditions the initial conditions,as calculated with the ciLtrim, are added to the control inputs. The turbulencegenerator block computes horizontal gusts and the gust induced angle of attack. Thismodel will be discussed in the next chapter. The resulting variables of a simulation arecopied into the Matlab workspace:

time A vector containing the time of each iteration step;output A 7 outputs matrix, one column for each output;Place The relative position of the aircraft at each iteration step;inputs The input signals calculated at each iteration step.

The block filters contains band pass filters to filter out noise and very low frequencies.Filtering out the low frequencies is rather crucial, because otherwise all pilot commandswould be considered as a disturbance and the command signals would be regulated tozero.

The inputs (ua) of the CITATION subsystem (Figure C.3 and Figure C.4) differ fromthe inputs of the linearization and trim model. The trim surface deflection angles aswell as the under carriage extended/retracted signal (u.c.) have become part of theinput vector. For the thrust control signals a separate vector is defined (Tx). The gustinputs are pulled out the input vector and are combined with the gust accelerationsand asymmetric gust inputs in a vector [Lust. The input vectors as described abovebecome:

ua

Tx

gust

massinit

[8e 8a 8r 8et 8at 8rt 8f U.c. 8sb ]T

[TXl TxJT

lUg Cig (3g Ug o.g ~g Ugasymm Cigasymm]T

(3.10a)

(3.10b)

(3.10c)

(3.10d)

where:

8e ,8a ,8r

8en 8an 8rt

8j, 8sb

U.c.

TXl , TX2

U g , Cig , (3g

Elevator, aileron and rudder deflection angleThe corresponding trim surface deflection anglesFlap and speedbrake deflection angleUndercarriage retracted or extended signalThrust input, engine 1 and 2Horizontal gust speed, gust induced angle ofattack and gust induced sideslip angle.Asymmetric components of gust

The states (;!:.), state derivatives (i) and outputs (~) used by the controller are:

£ = [p q r V Ci (3 4> () 'ljJ h x yf. dx=-x- dt-

Y - ['"'I flight path acceleration vert. acceleration ..."-<7.CC - I

... / azimuth angle bankanglef

(3.lla)

(3.llb)

(3.llc)

- 21 -...<.. Extensions of the non-linear model for direct lift controle----------~

In the following, a brief description of the systems surrounding the CITATION subsystemwill be given:

trim conditions In this block, the input values resulting from the trim_ac routine areadded to the control vector. When no controls are provided, the aircraft would flyaccording a straight and level flight path.

Turbulence Generator In this block, the horizontal gusts (ug ) and the gust inducedangle of attack (ag ) are calculated. A detailed description of the working of thisblock is outlined in chapter 4.

control selection The control signals used for controller design are selected out ofthe much larger control vector presented to the nonlinear model.

Actuator dynamics & limiters The commanded control signals generated by thecontroller are passed through a washout filter, followed by a first order actuatordynamics model. The control signals are limited as far as deflection and deflectionrate are concerned.

inner, flaps and speedbrakes loop Proportional feedback controllers for gust alle­viation. A detailed description of these controllers is outlined in chapter 4.

Filters Contains band pass filters to filter out noise and filter out very low frequencies.Filtering out the low frequencies is rather crucial, since otherwise all pilot com­mands would be seen as disturbance and the command signals would be regulatedto zero.

Sensors Not really sensors, but a calculation of the several output signals from thestates, state derivatives and flight path vector.

- 23 -~ Linear dynamics modelse --------------------------~

4 Linear dynamics models

To design a controller for aircraft attitude/altitude hold, it is expedient touse a linear aircraft dynamics model. In [Gerrits'94] a general model foraircraft motion has been derived. This model is divided into two parts:

1. Symmetric motions, in the symmetry plane of the aircraft. The sym­metry plane is the vertical plane through the centre of gravity andthrough the nose. Symmetric motions include for instance pitchingmotions, height change, etc.

2. Asymmetric motions, perpendicular to the symmetry plane. Asymmet­ric motions include for instance heading change, roll, etc.

In section 4.1, this model will be given and extended with the outputs neededfor DLC. The stability derivatives of the DLC control surfaces are added 0 themodel. In section 4.2 this model is extended with the horizontal and verticalwind speed derivatives. In section 4.3, the dynamics of the actuators, drivingthe control surfaces are derived from the gust bandwidth and the dynamicsof the actuators actually present on the aircraft. When these models are setup, the open loop system characteristics of the aircraft can be calculated.These characteristics are presented and discussed in section 4.4.

4.1 Citation state space model

For gust alleviation with direct lift control (DLC), only the symmetric motion will beconsidered, since DLC surfaces have most effect on motions parallel to the symmetryplane. For this research the vectors given in (4.1) will be used for a symmetric motioncontroller design. The disturbance (.4), input (.1£), state (.I.), measurement (~) and output

- 24 -Direct lift control for the Cessna Citation II.r'e --------------------------------

,)'

('l1J vectors concerning symmetric motions are given by:

51 lUg O:gjT

Y. [8e 81 8sb jT

~ [V 0: () qjT (4.1 )

~ [V 0: () q h It 4 I aZA 8e 81 8sb jT

'lL - [V 0: () q h It 4jT

where 51 is the gust input vector with inputs for horizontal gust and gust induced angleof attack. Vector y. is the control vector with inputs for the elevators, the flaps and thespeedbrakes. Vector ~ is the state vector, 'lL the state vector enlarged with vertical speed,vertical acceleration and pitch acceleration added. The vertical speed and accelerationcan be calculated from the states. The vector ~ contains all measured (calculated)variables and all control inputs. The model of symmetric motions will be built up littleby little, starting with a basic model for symmetric motions, with the elevator deflectionangle as only control input. After that, the model is extended with the control inputsfor the flaps and the speedbrakes. Finally, the model is extended with the inputs forthe horizontal gust speed and the gust induced angle of attack.

The basic aircnd't model for the symmetric motions with no gust influence and only theelevator command signal as input, has been derived in [Gerrits'94]:

CXa

CZa + (Cz" - 2flc)DcoCma + Cm"Dc

(4.2)

where Dc = -f-lr All other scalars in the matrix in (4.2) are constant. In the Laplacedomain equation (4.2) is:

(4.3)

where -f-s is the Laplace transform of Dc = -f-1t. The model is now recognized as statespace model with state and output vector'lL = ~ = [u 0: () qjT and input U = 8e :

;r A~+ By'(4.4)

'lL C~+ Dy'

with

A T- 1( -P1-

1 Pz)T (4.5a)

- 25 -Linear dynamics modelse -------------------------------

.)t:::.

B

C

D

(4.5b)

(4.5c)

(4.5d)

Matrix T is the transformation matrix from a non-dimensional model to a model withdimensions:

[

lOOO]T= 0100

001 0000.f..v

(4.6)

To use the DLC-surfaces as inputs, the linear model is extended with the stabilityderivatives of the flaps and the speedbrakes, as has already been done for the elevatorsin (4.2). The right-hand side of (4.2) becomes:

(4.7)

with corresponding input vector II = [8e 8f 8sb ]'

The state model is completed with the addition of the horizontal gust speed and the gustinduced angle of attack stability derivatives, discussed in detail in [Van der Vaart'93].The gust acting on the aircraft induces forces in forward (X) and downward (Z) directionand induces a moment (m) about the lateral axis. The wind speed in horizontal directionis U g and the gust induced angle of attack Cig are added to the state space model (4.2).The total state space model, including gust velocity effects, becomes:

[ Cx, - 2fl,D, CXa CZoCx, ] [U]CZu CZa + (Cz" - 2j1c)Dc -CXo CZq +2j1c . a

0 0 -Dc~mq - 2j1cJ(~Dc ~Cmu Cma + Cm"Dc 0 ---..-.-

:it. (4.8)CX6e CX6

Cx...] [8'] [Cx" Cx

• ]f

~z.: .[~ ]CZ6e CZ6 CZ6 8 CZuf .b • f - 9

0 0 o 8 0Cm6e Cm6 Cm6 ~ Cmug Cmag ~

f .b!!

whereCXug CXu CXag CXa

CZug CZu CZag CZa (4.9)

Cmug Cmu Cmag Cma

- 26 -~ Direct lift control for the Cessna Citation II~ --------------------------------.J'

(4.10)

This is the non-dimensional representation of the aircraft dynamics.model with dimensions, the following substitutions are used:

A T-1A.T

B T- 1i3E T-1i;

To convert to a

(4.11a)

(4.11b)

(4.11c)

with T the transformation matrix as given in (4.6).

For turbulence alleviation, which is a height hold system, output equations for verticalspeed, vertical velocity and pitch acceleration will be added to the output equations.The pitch acceleration is added for calculation of the vertical acceleration at a pointof measure other than the centre of gravity. All equations can be calculated fromthe available states or state derivatives of (4.2). With approximations sin, ~ , andcos, ~ 1 and substitution, = 0 - a (see Figure 2.1, page 10), there holds:

h Vsin, ~ V(O -a)

h = axsin, - azcos, ~ 11(0 - a) +V(0J ~ V(O - a) = az-y

(4.12a)

(4.12b)

It is assumed that ax ~ az, which is valid since, mostly, 11 ~ V. So, in a smallspace around the trimmed condition, the true airspeed is assumed to be constant inthe calculation of the vertical acceleration. However, when no auto-throttle is used, theairspeed will decrease due to DLC-surface deflection and h will be a bit larger than inreality. For the criterion (2.3) and (2.1) this is not very important. If the controller canminimize these criteria, it performs slightly better in reality.

With substitution iJ = q and J2. = [0 - V V 0], equation (4.12a) and (4.12b) can bewritten as function of the states and the state matrices of (4.2):

h

h

q [OOOI]i

J2. (A£ +Bli)

[0 0 0 1] (A£ +Bli)

(4.13)

- 27 -~ Linear dynamics models<eJ -------------------------------~

With the extension of the output vector with vertical speed, vertical acceleration andpitch acceleration, the output vector becomes lL = [V 0: () q h 'it q]. Equations (4.13)are added to the C and D matrix of (4.4) by adding three row-vectors. The C and Dmatrix are:

1 0 0 00 1 0 00 0 1 0

C 0 0 0 1

12.12.A

[000 1] A

0

D = o 0 0 0 012.(E IB)

[OOOl](EIB)

(4.14)

(4.15)

The gust and control matrices Band E have been merged to one big matrix (E I B)for simplicity.

d

u

aircraftmodel

controller ~

z

y

Figure 4.1: General model representation

The general state space model is drawn in Figure 4.1, with state space equations:

Ar + (E I B)[~]

C;r+D [~](4.16)

The disturbance (d.), input (li), state (;r), measurement (.~) and output (lL) vector are

- 28 -Direct lift control for the Cessna Citation II"..,

~ ---------------------

given by:

lUg agV[8e 8f 8sbV[Va eqV[V a eq h h <i I aZA 8e 8f 8sbV[Va eq h h <iV

(4.17)

The flight path angle I in ~ is calculated by I = e- a and the vertical accelerationat location XA relative to the centre of gravity is calculated by aZA = aZcg - xAq. Thelinear model can be derived from the nonlinear model, as represented in Simulink, bythe Matlab routine 'LINMOD'. The result of the linearization routine is a twelve statelinear model representation of the nonlinear model (section 3.3) including symmetricand asymmetric motions. This model is valid for small disturbances about a trimmedoperating point (~,1~:()). For controller design, only the states representing the sym­metrical motions are of interest. So, all other states are set to zero, leaving [u a e q]as states and [V a e q h h q] as outputs. The matrices of the complete twelve statemodel are (reordered) presented in table A.1 for a linearization with V = 166 m 8-1 andh = 5000 m. The upper left six by six matrix represents the symmetric motions andthe lower right six by six matrix the asymmetric motions. The states of the reducedlinear model do not include the height and relative x position of the aircraft. Omissionof these states is valid, because no other states are dependent on height and x position.The remaining parts are cross coupling terms, which are small enough to be neglected.The state space model can be divided into four submodels: one for symmetric motions,one for asymmetric motions and two for the cross coupling terms. All models are of theform:

" _{;!:.l = AU K1 +BU 1l1~symm - C +DlL1 = uK1 u1l1

(4.18)

The models for the asymmetric motions and the cross coupling terms can be calculatedlikewise. The asymmetric model has index '22'. The cross coupling term (~crosst)

representing the effect on the symmetric motions as a result of a deflection of asymmetricmotion controls is denoted with index '12' and the other cross coupling term (~cross2)

is denoted with index '21'. Neglecting the cross coupling terms is valid, because themaximum transfer between one of the inputs and one of the outputs of the cross couplingsystems is neglectable compared to the maximum transfer of the symmetric system, thus

II~crosstiloo <t:: II~symmlloo and II~cross21loo <t:: II~symmlloo' The infinity norm is calculatedby:

11~lloo = max o-(~(jw))w

(4.19)

where 0- is the maximum singular value of the transfer function ~. For the system given

- 29 -..:r:::::.. Linear dynamics models€;----------,)'

in Appendix A there holds:

IIE symm \100IIEasymm 1100

II Ecrossl 1100II Ecross21100

157878642634

(4.20)

which shows that neglecting the cross coupling terms is valid.

The output vector of the reduced model is not exactly [u a () q hhq]T. The horizontalspeed u is approximated by the true speed V. This approximation is valid, since u islinearized as u ~ V - 1.82a + 0.0611 with 1.82a « V and 0.06 11 « V when a, 11 <±20deg.The same holds for h, which is calculated as h= V(-a+())+0.064> and is approximatedby h = V(-a + ()) since 0.064> « V( -a + ()). Adding h to the output vector does notmodify the state and input matrices (A and B). So it can be added safely to the system.The same arguments hold for hand q, which are calculated by (4.13).

RemarkThe influence of the speedbrakes on the system is questionable. The stability derivativesZOsb = -0.004 and XOs b = 0.872 in table A are much smaller than the stability derivativesof the elevators and the flaps. Perhaps something went wrong with the implementationof the speedbrakes aerodynamics in the nonlinear citation model, although the modeldata is copied literally from the Cessna Citation 500 model of [Broos'87].

4.2 Turbulence state space model

(4.21a)

(4.21b)SWgWg(W)

To simulate the influences of turbulence on aircraft, it is necessary to have a suitablemodel for this turbulence. There are several models for calculating turbulence signals.The one that will be used here, uses the Dryden spectral form for calculation of theturbulence signals from a white noise source. The Dryden spectral densities in horizontaland vertical direction respectively are given by [Van der Vaart'93]:

S () 2,.,.2 Lg 1

UgUg W = v V 1 + (Lrr2L9 1 + 3 (L~Wr

(7 - -----'----'-n

V (1+ (¥rrwith (72 the variance of the gust velocities. The positive directions of the horizontalgust component ug and the vertical gust component wg are chosen along the negativestability reference frame axes (see Figure 2.1). The spectrum for gust in vertical directionis shown in Figure 4.1a and the spectrum in horizontal direction is given in Figure 4.1b.In both figures, the spectra are drawn for gust wavelength Lg = 150m. and Lg = 1500m.The spectrum with L g = 150m. having the biggest bandwidth.

- 30 -A. Direct lift control for the Cessna Citation II

€,------------~

VertICal DrydBn spedrum, Lg_l50 and 1500m

10'frequency

(a) vertical direction

~j 10·

1r···";....,..",. ,.',c,,:, : ,C), ,,~, ·co ..:''':'':':':",':, ...... :· ";':',: 'lJ'

~~

~ ,0·VI H'!/:nC:!!:\[.~ ••:i Hi·!'!1

'0'frequ..cy

(b) horizontal direction

Figure 4.2: Dryden spectral densities

Gust can be seen as a pattern frozen at a fixed point in the air, representing the gustspeed, with wavelength Lg and with normally distributed amplitude with variance a.The aircraft is flying through this gust pattern with speed Y. The Dryden filter isformed by:

(4.22)

A white noise input signal is chosen with spectral density Syy(w) = 1. If the horizontaland vertical gust velocities U g and W g are considered as outputs of the filter, then thefilter functions are defined as:

(4.23a)

(4.23b)

with WI and W3 white noise sources. Solving (4.23a) and yields, with substitution jw = s,the following frequency response functions:

J2Lg 1HUgW1 (s) = a -Y L1 +::::.9..sV

H () _ ~g 1 +V3~sWgW3 S - a Y ( L) 2

1 +::::.9.. sV

(4.24a)

(4.24b)

With substitutil'il of s = ddt' the frequency response functions can be written in a general

- 31 -~ Linear dynamics models~ ----------------------------------~

state space description:

Ug = (-~) U g + (augfE) WI (4.25)

wg +~ ?Dg + (~) 2 Wg = ,,~ (~) 3w,+ "J~~?D3 (4.26)

With the substitution w; = wg - awgj¥;, the state space model for the vertical turbu­lence velocity can be written as:

(4.27)

where WI and W3 represent white noise sources. The state space model represents theDryden filter to calculate turbulence velocities from the white noise sources.

The velocity (V) of the aircraft's centre of gravity relative to earth can be expressed ina term L, the velocity relative to air and V g , the velocity of the air relative to earth(the gust velocity). The aircraft velocity becomes: V = L + V g , with magnitude V.The angle between the V and L is called the gust induced angle of attack a g , whichmay be approximated by

Wga g = -

Vfor small values of ug and W g •

In general notation:i g = Ag;£g +Bglig

'!!...g = Cg;£g + Dglig

with1 0

~ ]0 0

0 1 0 0Cg = v 0 Dg = /¥; 0- L g aUg Lg

0 0 0 aj¥;Cig L g

The state, input and output vectors are defined as follows:

(4.28)

(4.29)

(4.30)

(4.31 )

iJ = [ ug a g a; ]yT = [ ug a g ug Og ]-g

li; = [ WI W3 ]

where WI and W3 are independent white noise sources.Matrices Cg and D g are used for calculating the time derivatives of the gust velocity.The Simulink model for gust simulation is shown in Figure C.13. The two boxes onthe left side represent the white noise sources and the big block in the middle the statespace model. The multiplexer and demultiplexer are used for signal selection.

- 32 -~ Direct lift control for the Cessna Citation II

e-----------------~

4.3 Actuator dynamics and washout filter

(4.33)

(4.32)

(4.36a)

(4.36b)

A very common design approach is to approximate the actuator behaviour with a firstorder system, as is done in [Bogers'93]. The actuator for the elevator is a simple firstorder model:

8 K a

8 w 1+ TaS

with 8 the real deflection angle, 8w the deflection angle at the input of the actuator, Ka

the actuator gain and Ta the actuator time constant.

A washout filter is added to the controller to overcome steady deflection angles. With thewashout filter active, the flaps and speedbrakes will return to their neutral points aftersome time. After this time they can be used with full deflection range. This washoutfilter should be active for gust alleviation only, otherwise, step commands would beignored, leading to dangerous situations.

The system of the actuator preceded by the washout filter is represented by:

8 TnS K a

8c 1 + TwS 1 +TaS'-..--'" '---vo-'

washout actuator

with 8c the commanded deflection angle generated by the controller, Tn the washoutnumerator time constant and Tw the washout time constant. In the time domain thesystem is given by:

u. •T18 +T28 +8 = Ka Tn 8 c (4.34)

with T1 = TaTw and T2 = Ta + Tw . Using 8* = 8 and integrating (4.34), the actuatorsystem is described by:

with ;KT = [8 8*].

In Figure D.3, the step response of the washout filter is plotted for different values ofthe washout filter time constant. When a constant deflection is commanded, the controlsurface would stay in at a constant deflection angle, not able to be used in full rangefor further gust alleviation. With the washout filter, the control surface goes back to itsneutral position after some time. This process is shown in Figure D.3. For the controllerto be designed, the time constant of the washout filter is chosen to be Tw = 6 sec.

The DLC surfaces are to be operated about a neutral point, from which they can movein positive and negative direction, following the commanded deflection. The speed­brake effectiveness is a measure for the overall influence of a speedbrake deflection.The speedbrake neutral angle is chosen halfway the second line segment in Figure B.6,corresponding to 8SB = 0.64 rad. For the flaps, the neutral point is chosen halfwaythe maximum deflection angle: 8F = 0.34 rad. The absolute deflection angle of thespeedbrakes and the flaps is given by:

8SB

8F

- 33 -~ Linear dynamics models

~ --------------------------------~

surface 8min (deg) 8max (deg) 8max (degjs) 8neutral (deg)

elevator -18.8 15.7 0flaps -20 20 5.7 20speedbrakes -20 20 60 37.5

Table 4.1: Minimum and maximum deflection rates and angles

with 8SBo the neutral speedbrake deflection angle and 8Fo the neutral flap deflectionangle. In the remainder of this report, only deflections relative to the neutral angle willbe considered.

The required maximum DLC deflection rate 8DLmax for ideal gust compensation is pro­portional to the maximum DLC deflection and the highest efficient gust frequency Wg

[Schanzer] :(4.37)

The gust frequency depends on the airspeed and the minimum gust wave length Amin:

211"wg = --V

Amin(4.38)

The shortest effective gust wave length is about 0.3 of the characteristical gust wave­length: Amin ~ 0.3Lg • For the initial investigation simple first-order approximationsof the elevator and DLC (flaps and speedbrakes) were used. A block schemetic of thesystem is shown in Figure C.14.

For the maximum deflection angle and maximum deflection rate, there holds:

18Ddt) 1$ 8DLmax

8DLmin $ 8 DL $ 8DLmax

(4.39a)

(4.39b)

An overview of the minimum and maximum angles and rates is given in table 4.1:

The dynamics of the elevator actuator are represented by the first order system

(4.40)

where 8ec is the commanded elevator deflection generated by the controller, J{e the gainfactor of the dynamics system and Te the time delay. With J{e = 1 and Te = 11

3(see

[Bogers'93]). The time constants of the flaps and speedbrakes actuator will be derivedfrom the open loop gust bandwidth in section 4.4. The entire linear system includingthe sensor equations, the actuator dynamics and three blocks reserved for the controllersis presented in Figure C.15. The following systems are included in the model:

- 34 -

~ Direct lift control for the Cessna Citation II@9----------~

Turbulence The turbulence generator. The horizontal gust velocity U g and the gustinduced angle of attack O'.g are calculated with the state space model derived insection 4.2.

Controls This block contains one 'scope' per input signal. With these scopes thecontrol signals can be watched during a simulation.

State Space model Citation II In this block, the state space model is calculated asderived in section 4.1, equation (4.16).

Video wall 'scopes' for the outputs of the state space model.

Calculate place Calculates the x-position of the centre of gravity relative to the placeat the beginning of a simulation.

Filters Contains band pass filters to filter out noise and filter out very low frequencies.Filtering out the low frequencies is rather crucial, since otherwise all pilot com­mands would be seen as disturbance and the command signals would be regulatedto zero.

Inner, flaps and speedbrakes loop The controller for the elevators, the flaps andthe speedbrakes respectively (see also Figure 2.2).

Actuator dynamics & limiters This block contains the washout filters, the actuatordynamics and the deflection rate and deflection angle limiters.

4.4 Open loop system characteristics

To see the improvements of the system response when the controller is active, theopen loop characteristics of a system have to be known. For one trim condition thesecharacteristics are drawn in Appendix D. The trim-condition for which these plots aregenerated is:

h 5000mV 166 ms-1

The plots are generated using the linear system from (4.29). In (D.1) the open loopsystem poles are plotted in the complex plane. The complex pole pair at great distancefrom the real axis represents the short period oscillatory mode and the complex polepair near the origin represents the phugoid mode (Figure D.2). The remaining poles onthe real axis are the time-constants of the actuator dynamics and the washout filter.The damped natural frequency and damping ratio are displayed in table 4.2:

Figure D.4 shows the bode-plot from white noise input to horizontal gust speed (ug ) andgust induced angle of attack (a). This plot is drawn for gust wavelength L g = 1500 m.The bandwidth of gust with this gust wavelength is 0.2 rad S-1. For a much shortergust wavelength of L g = 150 m, which is also very likely to appear, the bandwidth is 2

- 35 -~ Linear dynamics models

@---------­,JIC.

Phugoid modeShort period modeWash out filterElevator actuatorSpeedbrake actuatorFlaps actuator

Pole location-0.0134±0.0835j-1.9693±4.1469j

-0.1667-13-20-20

Damping0.15880.4290

1111

Natural frequency0.08464.59000.1667

132020

Table 4.2: Open loop system poles

rad 8-1

• Therefore, the time constant of the actuator dynamics of the DLC surfaces ischosen:

1Tf = Tsb = 20 8 (4.41 )

being about ten times smaller than the time constant of gust, so the actuator can operatein the complete gust bandwidth. The time constant of the elevator actuator is T e = 113s,according to [Bogers'93].

Gust bandwidth

The bodeplots of horizontal gust speed (ug ) to vertical speed and acceleration (FigureD.5) show that U g influences it mostly, with -3 dB gain bandwidth at about 1 rad 8-

1.

The gust induced angle of attack has much more influence of both it and h. Therefore,it is necessary to create a DLC controller for these motions. The pitch acceleration qis only altered by the gust induced angle of attack at high frequencies, but cannot becontrolled by DLC.

Elevator effectiveness

From bode plots Figure D.7 it can be concluded that the elevators can be used to controlthe phugoid mode when feeding back the pitch angle signal. This is useful to compensatefor a change in pitch angle induced by ago Figure D.8 shows that the elevator can beused to augment stability for the phugoid mode as well as the short period mode by apitch rate feedback. From Figure D.9 it can be concluded that the transfer from elevatorto vertical speed has a very bad phase response, which creates a delay in lift build upas explained in chapter 2.

- 36 -~ Direct lift control for the Cessna Citation II

S----------~

Flap effectiveness

Figure D.10 shows the drastic increase in drag due to a flap deflection. If the flaps areused for vertical acceleration improvement solely, the change in forward speed shouldbe compensated by other means. It may be clear from this figure that only very smallpitching moments are introduced be the flaps. Figure D.ll proves the very small in­fluence of the flaps on angle of attack and pitch rate. Fortunately, the influence ofthe flaps on the vertical speed is enormous, although the transfer from flaps to verticalacceleration is about unity. No big pitch accelerations are introduced by the flaps.

Speedbrake effectiveness

The drag introduced by the speedbrakes is roughly the same as introduced by theflaps, as can be seen by comparing Figure D.13 to Figure D.lO. The transfer fromthe speedbrakes to a, () and q is, as expected, very small. The transfer to verticalacceleration is not very large, while a larger magnitude was to be expected.

- 37 -~ PID controller design

@9----------~

5 PID controller design

In this chapter, the feedback gains for the elevators (section 5.1), the flaps(section 5.2) and the speedbrakes (section 5.3) will be determined from theroot locus plots. In section 5.4 the results will be discussed and interpreted.

5.1 Elevator control

For the short period motion, the handling qualities are of great account. For goodhandling qualities, the damping ratio of the short period mode should be (sp > 0.6and the damped natural frequency should be W n > 6, which is shown in Figure 5.1[McLean'90].

The point marked 'desired' shows the required minimum handling qualities and thepoint marked 'Citation' shows the handling qualities of the uncontrolled Citation IIaircraft. It can be seen that the damping ratio as well as the natural frequency shouldbe increased. This can be achieved by proper feedback of output variables to the controlsurface deflections.

Of cource, the handling qualities cannot be optimized at the cost of the ride discomfortindex as described in section 2.1. With excellent handling qualities the gust alleviationperformance degrades. The reverse, if gusts have to be alleviated at all cost, the han­dling qualities will be poor. Optimizing an RDI can be done by minimizing verticalacceleration and pitch acceleration. The controllers are designed for minimizing handh. If t1h = 0, then h= O.

The following sections describe the feedback of the several output variables to the controlsurfaces used with DLC, i.e. the flaps and the speedbrakes, and the elevator as discussedin chapter 2. The controller structure is shown in Figure 5.2:

The remainder of this chapter is based on the assumption that a negative outputfeedback is used:

1J.=-klL

so, a negative value for k means positive feedback!

(5.1)

- 38 -~ Direct lift control for the Cessna Citation II@J----------~

0.6

0.5

0.4

0.3

0.2

0.1

X desired

X Citation

'---+--+---+--+--I---+--- W sp

1 2 3 4 5 6Figure 5.1: Handling qualities

gust

aircraftmodel

comfortcriteriaoutput

(8, q)

(Ii,ii,ct)

Figure 5.2: Controller structure

The elevator is less appropriate for gust alleviation, because of the non-minimum phaseresponse in a feedback loop with the vertical velocity h. This means that the changein vertical speed is in first instance in the wrong direction and later on in the desireddirection, causing the elevators to oscillate when proportional feedback is used. Thisresults in an unstable pole when the feedback gain is increased. When, for example, the

- 39 -~ PID controller designe----------~

vertical velocity increases, the elevators are moved down (positive deflection), resultingin an increase in lift, followed by a decrease in lift. Only the decrease in lift is desired.In the root locus plot of elevator and vertical speed this can be seen from the slow polesmoving directly into the right half-plane. Therefore, the elevators will be used mainlyfor attitude control and the DLC surfaces for altitude control. For attitude control thepitch angle (e) and pitch rate (q) are fed back to the elevators with a PID controller. Acontrol loop for the DLC surfaces will take care of altitude control. The output signals,which are fed to the controllers, are obtained from a rate gyro and a rate integratinggyro, which measure pitch rate and pitch angle respectively. As mentioned in chapter 2,the elevators are less appropriate for gust alleviation control, because of the delay in liftbuildup after an elevator deflection. Figure D.9 illustrates this effect. The transfer fromDe to it shows in the controller bandwidth a phase angle of about -270 degrees, which isa 90 degrees phase arrears. The elevator deflection needed for gust alleviation will be afraction of the commanded flap deflection signal, which is mainly meant for pitch angleand pitch rate corrections. The controller for the elevator is presented in Figure C.16.

Pitch angle to elevator feedback

As can be seen in Figure D.7, the elevator will mostly affect the phugoid flight modewhen it is used in a feedback loop with the pitch angle e. The sign of the e to Defeedback gain I<eeis easily derived: When the pitch angle increases the aircraft makes anose-up motion, which has to be corrected by a positive elevator deflection. Therefore,I<eeshould be positive. In Figure E.1 and Figure E.2 the root locus plot for increasinggain I<eeis plotted (Figure E.2 shows the phugoid mode). Since the elevators mostlyaffect the phugoid mode, in the first instance the phugoid mode will be considered. Forincreasing I<ee the relative damping and absolute damping both increase. From the pointof view of handling qualities, a feedback gain at the point where the relative damping( = 0.8 would be a good choice. The gains found for various relative damping ratios arelisted in table E.1. The gains are marked with a + sign in the root locus plot. FromFigure E.1 it may be concluded that increasing I<eemakes the short period dampingonly slightly worse. The gain for pitch angle to elevator deflection I<eeis found to be

I<ee = -0.10

Pitch rate to elevator feedback

(5.2)

The same reasoning as done for I<ee can be done for I<eq. From Figure D.8 it canbe concluded that a feedback of the pitch rate affects both phugoid and short periodmode. The sign of the feedback gain I<eqshould be the same as for pitch angle feedback,because, if q is positive, the aircraft is tumbling backwards. This should be corrected bya positive elevator deflection. Therefore, I<eqshould be positive. The root locus plots forincreasing I<eq, Figure E.3 and Figure EA, show that the best results can be gatheredat the short period mode, by that affecting the phugoid mode only a little. According

- 40 -~ Direct lift control for the Cessna Citation II

~--------.J'

poles ( Wn

-0.0750±0.0563j 0.7998 0.0937-3.9408±3.5321j 0.7447 5.2920

Table 5.1: System poles when inner loop closed

to Figure 5.1, the relative damping of the short period mode should be at least ( = 0.6with natural frequency W n > 6. For the short period mode, the gain factor that fits bestto the handling requirements is calculated at ]{eq = -0.11 A proper choice of ]{eq thatsatisfies this requirement fairly, should be

]{eq = -0.15 (5.3)

table E.2 shows the gain factors for other damping ratios. The feedback gains ]{e(J and]{eq together form the attitude hold controller. An additional feedback gain is addedto correct for the pitching moment introduced by the speedbrakes and the flaps. Theinput signal for this gain is obtained from the flap deflection command signal. Theclosed loop system poles, are given in table 5.1. The short period system poles areplotted in a pole-zero map in Figure E.5 and the phugoid poles in Figure E.6.

5.2 Combined elevator/flap control

With the specifications as given in section 2.3, the flaps can be used for gust alleviation.No feedback of attitude variables will be used, only vertical speed and acceleration andpitch acceleration will be used for feedback. Pitch acceleration is fed back for minimizingthe vertical acceleration at the pilot's seat, which can be calculated by (2.8). Thecontroller structure for the flaps is presented in Figure C.17.

Vertical speed to flaps feedback

The bode plot of the transfer from flaps to vertical speed (Figure D.12) shows thatthe phugoid mode is affected at most. The sign of the feedback gain ]{fJollows from

the fact that the aircraft is moving up (positive h), the flaps should be retracted, bythat decreasing the total lift. As a result, the aircraft will loose height. For positive hthere should be a negative 8f . Figure E.7 shows the root locus plot for decreasing ]{fi-.'

Table table EA shows the gains for several relative damping ratios. A proper choice for]{fi-. should be

(504)

- 41 -~ PID controller design~ -------------------------------~

poles ( Wn

-0.0972±0.0464j 0.9025 0.1077-6.2446±4.1303j 0.8336 7.4913

Table 5.2: System poles when inner loop closed

Vertical acceleration to flaps feedback

When the aircraft is accelerating upwards, the flaps should be retracted, so, as withvertical speed feedback, there should be negative feedback. For the short period os­cillatory mode, the gain factor that fits best to the handling qualities requirements iscalculated at Kf;.. = 0.224. For a trade off between optimum performance and handlingqualities, this gain was selected. As can be seen from the root locus plot for decreasingKf;.. (Figure E.8), this should be a fair choice. So, the selected gain is

Kf;.. = 0.224

Pitch acceleration to flaps feedback

(5.5)

The third feedback gain K fq , from pitch acceleration to flaps, is found in the same way asbefore. The bode plot Figure D.12 shows that the flaps will affect the pitch accelerationmost at higher frequencies, so the short period mode is to be examined. The sign ofthe feedback gain is positive. If the aircraft is accelerating backwards, this should becompensated by a positive flap deflection. With the flap deflection angle positive, theflaps induce drag aft of the centre of gravity, by that creating a negative pitch rate.Root locus analysis (Figure E.9) shows that only small feedback gains would be correct.The feedback gain is chosen:

K fq = 0.12 (5.6)

The system poles after closing the loop are given in table 5.2.

To operate the flaps between their deflection limits and within the margin for deflectionrate, the commanded deflection is reduced to practical levels. This is done by alleviatingthe commanded flap deflection angle.

5.3 Cornbined elevator/ speedbrake control

Vertical speed to speedbrakes feedback

When the aircraft is moving upwards (positive h), the speedbrakes should be extended,by that increasing the total drag and decreasing the lift/drag ratio. As a result, the

- 42 -Direct lift control for the Cessna Citation II/'e ----------------------

~

poles ( Wn

-0.1088±0.0639j 0.8624 0.1262-5.7438±2.2588j 0.9306 6.1720

Table 5.3: System poles when inner loop closed

aircraft will loose height. Thus, the feedback from vertical speed to speedbrakes shouldbe positive. Bode plot Figure D.15 shows that the speedbrakes are most effective forthe phugoid mode, as far. as vertical speed control is concerned. A very small gain isselected for feeding back h to the speedbrakes:

]{sbi. = -0.08

Vertical acceleration to speedbrakes feedback

(5.7)

The same reasoning as with vertical acceleration to flaps feedback is done. The speed­brakes are working in opposite direction as the flaps, so a positive feedback gain shouldbe used. The bode plot in Figure D.15 shows that the phugoid mode is to be examined.In Figure E.I0 the root locus plot for the phugoid mode for increasing ]{sbj, is drawn.The gain corresponding to a relative damping ratio ( = 0.8 was chosen for feedback,this gain is:

]{sbj, = -0.93

Pitch acceleration to speedbrakes feedback

(5.8)

At last, a gain for pitch acceleration to speedhrakes will be selected. From the bodeplot it can be concluded that the short period rnode is affected at most. The sign of thefeedback parameter is negative. When the aircraft is accelerating backwards, a pitchrate opposing this motion should be generated. When the speedbrakes are extended,drag is induced above the wings, creating a positive pitch rate. Therefore, if the aircraftis tumbling backwards, the speedbrakes should be retracted, by that decreasing the dragabove the wing. The root locus plot for decreasing ]{sb,)S drawn in Figure E.l1. Also,the gain that fits best to the handling qualities requirements is calculated. While thisgain gives good results in simulations, it is selected as the feedback gain:

]{sbq = 1.89

The locations of the closed loop system poles are given in table 5.3.

(5.9)

- 43 -~ PID controller design

~ ----------------------------------~

5.4 Simulation results

Because the trim routine (section 3.2) does not work correctly, the results of nonlinearsimulations are not presented in this report. After trimming the aircraft, the openloop response of the nonlinear system is not a straight and level flight. Low frequencyoscillations appear on almost all states.

The controllers, designed in the preceding sections, are tested with different flight con­ditions. For one condition, height 5000 m. and speed Mach 0.5, the impulse responsesof the open loop and the closed loop system are compared. Figure E.15 shows theimprovement in impulse response for pitch angle to elevator feedback. An impulse isput on the elevator deflection command signal and the resulting pitch angle is drawnversus time. The big oscillation represents the uncontrolled response. With pitch anglefeedback, the response is no longer oscillatory and there is only little overshoot. The(short period) pitch rate response (Figure E.16) is improved less that the pitch angleresponse, but it is also a non oscillatory response. The flaps perform well on the verticalspeed response. There is no overshoot and there are hardly any oscillations. The resul­ting vertical accelerations are not good. The initial response is worse compared to theresults with the inner loop controller. The speedbrakes perform well on both verticalspeed and pitch acceleration. These results are almost as expected in chapter 404.

The results of the simulations with a filtered white noise input look promising. Theresponse of the states as a function of time is plotted in Figure F.1 and Figure F.2.These results are acquired using the elevator controller only. Figure F.3 shows therequired elevator deflection angle as a function of time. Figure FA. .. Figure F.6 showthe responses of the states and inputs with all controllers active. When the peak valuesof the signals in Figure F.1 are compared with the peak values of the signals in FigureF.4, it can be seen that the horizontal speed is worsened by a factor 2, the angle ofattack is improved by a factor 2, the pitch angle is improved by a factor 5 and thepitch rate remains roughly the same. Comparing Figure F.2 with Figure F.5 yields animprovement of the vertical speed by a factor 7, improvement of the vertical accelerationby a factor 2, improvement of the flight path angle by a factor 7 and improvement ofthe vertical acceleration at the pilot's seat by a factor 1.7.

From Figure F.3 and Figure F.6 it can be seen that the control surfaces never reachtheir maximum values as specified in section 2.3.

A comparison of the results of the simulations at several operating points is presentedin table F.1 up to table F.6. For each simulation, the root mean squared (r.m.s.) valueof the states and the inputs are calculated. In table F.1, the reference values of thesesimulations are given. These reference values are generated with a simulation with theinner loop (only attitude control) active. In table F.2 these values are set to 100 percent.The remaining numbers in table F.2 show the relative improvement or worsening of theseveral states compared to the results of the inner loop controller. The rows in the tablerepresent the following controller modes:

no controller the aircraft is not controlled

- 44 -~ Direct lift control for the Cessna Citation II@9----------~

inner loop

elevator loop

the elevators are used for attitude correction

the elevators are used for gust alleviation too

flaps+elevator the flaps are used for gust alleviation and the elevators forpitching moment corrections and attitude control

speedbrakes+elev. the speedbrakes are used for gust alleviation and the elevatorsfor pitching moment corrections and attitude control

DLC+elevator the flaps and the speedbrakes are used for gust alleviationand the elevator for pitching moment corrections and attitudecontrol

The rows of the table represent some interesting output variables:

~ horizontal speeda angle of attack() pitch angleq pitch rateh vertical speedh vertical accelerationq pitch accelerationI flight path angleaZA vertical acceleration at the pilot seat

The first simulation is done with the initial condition h = 5000m. and V = 0.4 Mach.The effect of the control surfaces on horizontal speed is enormous. The control surfaceactivity has a worsening effect on the horizontal speed for all controller combinations.Using the flaps increases the drag, because of the size of flaps. The speedbrakes do nothave much influence. Perhaps due to a modelling error (see section 4.4). From tableF.2 it can be seen that elevator control is a moment control technique. q and q, bothrelated to rotational motion, are worse than using the elevators for attitude control only.To improve vertical motion (h and h), attitude control is offered. When the flaps areused in combination with the elevators, the flaps are used for controlling the verticalmotion. The flaps are used for controlling vertical motion. The elevators are used forattitude control and correction for the pitching moment introduced by the flaps. Theresults of the combined elevator/flaps loop is good. An improvement by a factor 2 invertical acceleration and a factor 6 in vertical speed can be obtained. The flight pathangle remains more constant compared to attitude control only. The same hold for aand (). Hence, there is a decoupling between altitude and pitch angle. The effect ofthe speedbrakes in combination with the elevators or in combination with both flapsand elevators is neglectable. The ride discomfort index is reduced by more than onepoint. The same analysis are made on simulations at height h = 5000m. and speed~ = 0.5 Mach and ~ = 0.6 Mach. The results of these simulations are slightly betterthan the results discussed above. The control surfaces have more influence in controllingthe aircraft at higher speeds. table F.3 shows the control activity of the three controlsurfaces for each controller combination.

- 45 -~ PIn controller design

@9----------~

In table FA, the reference values of two simulations at height h = 10km. and onesimulation at h = 1000m. are presented. table F.5 shows the RMS values relative to theinner loop controller. The results are somewhat worse than the results of the simulationat h = 5000m. due to a different air density. At height h = 5000m. and speed vt = 0.3Mach. the results are worse than the results of the simulation at h = 5000m. due tothe low speed. So, the effect of the controller is better at higher speeds and the effect isworse at higher altitudes.

- 47-~ Conclusions & Recommendations@y----------~

6 Conclusions & Recommendations

Conclusions

The nonlinear model of the Cessna Citation II, present in Simulink, is modified andextended. Blocks for calculation of drag, lift and moment change due to speedbrakedeflection are added to the model. The aerodynamics of the flaps were already includedin the nonlinear model.

The structural modes of the aircraft, which are not considered in this report, should notbe neglected. Some modes are within the frequency range of the DLC controller system.

For the controller a decentralized structure has been chosen. This structure providesmeans for comparing combined elevator/flaps control and combined elevator speedbrakescontrol, as well as combined elevator/flaps/speedbrakes control. The feedback parame­ters are chosen using root locus design. Only proportional feedback has been considered,since with proportional feedback a good indication on the possibilities of DLC can beobtained. The controller for the elevator consists of an attitude hold controller and agust alleviation controller, which is derived from the flap control signal. For attitudecontrol only the pitch angle and the pitch rate are fed back to the elevator. The con­trollers for flaps and speedbrakes use the vertical speed, vertical acceleration and pitchacceleration for feedback.

The dynamics of the DLC actuators are estimated from the gust bandwidth. When theactuators have at least the same bandwidth, they can be operated for all gusts reachingthe aircraft. A washout filter is added to the controller, making sure the DLC surfacesare full range available after some time and do not stay at a certain deflection angleother than the neutral angle.

The controller for the elevator and the flaps turned out to be effective for gust alleviation,as was proved with the results of simulations for various flight conditions. Although thiscontroller is far from optimal, the results are very good. The speedbrakes turned out tobe less appropriate for gust alleviation.

- 48 -~ Direct lift control for the Cessna Citation IIe ------"-----------~

Recommendations

The speedbrakes turned out to be less appropriate for gust alleviation. Therefore, thefollowing classification could be made for the control surfaces:

1. The speedbrakes can be used for horizontal (drag) control. The big changes inhorizontal speed may be reduced using the speedbrakes.

2. The flaps are used for vertical flight path control, as is done in this report.

3. The elevators are used for moment control solely.

Furthermore, the left and right wing flaps can be operated separately, making asym­metric control possible. Gust will never attack the wing over the entire width homo­geneously, but different gust speeds shall be measured along the wing span. When theflaps are divided into left and right wing flaps, the contribution of the flaps to the lift,drag and moment coefficients becomes:

(6.1)

with ,6.C:nDL

the down-wash lag effect.

The last recommendation is considering the position of the sensors in the aircraft. Fornow, they are thought in the centre of gravity, but in the future realistic locations shouldbe determined.

- 49 -~ Bibliographye------------~

Bibliography

[Bogers'93] Bogers, R.A.E.M.Implementatie van een niet-lineair model van de Cessna Citation 500 in SIMULINKen het ontwerp van routines voor het trimmen en lineariseren van dit modelMaster's Thesis, Delft, 1993, Delft University of Technology

[Broos'87] Broos, P.M.Final mathematical models of the RLS Cessna Citation 500 aircraftPart III: Aerodynamics modelNLR TR 87091C, Amsterdam, 1987, National Aerospace Laboratory

[Damen'90] Damen, A.A.H. and Ven, H.H. van deModerne regeltechniek, dictaat 5662Eindhoven, 1990, Eindhoven University of Technology

[Erkelens'75] Erkelens, 1.J.J. and J.SchuringInvestigation on a passenger ride-comfort improvement system with limited controlsurface actuator performance for a flexible aircraftNLR VS-74-00l, Amsterdam, 1974, National Aerospace Laboratory

[Erkelens'74] Erkelens, 1.J.J. and J.SchuringPreliminary investigation on the ride-comfort improvement of a low wing-loadingtransport, including a limited comfort criteria studyNLR VS-74-00l, Amsterdam, 1974, National Aerospace Laboratory

[Flight '87] NLR, Flight Simulation GroupAircraft model for the multi cockpit simulator of eurocontrol experimental centre(MCS)NLR TR 87148L, Amsterdam, 1987, National Aerospace Laboratory

[Fry' 79] Fry, D.E. and Winter, J.S.The design of aircraft automatic ride-smoothing systems using direct-lift controlNLR TR 79045, Amsterdam, 1979, National Aerospace Laboratory

[Gerlach'81] Gerlach, a.H.Vliegeigenschappen I, deel I en II, Dictaat D26Delft, 1981, Delft University of Technology

- 50 -Direct lift control for the Cessna Citation II,./e -----------------------

~

[Gerrits'94] Gerrits, M.Direct lift control for the Cessna Citation II, A FC/NFT, a literature studyInternal NLR report NLR IW-94-021, Amsterdam, 1994, National Aerospace Labo­ratory

[JAR/AWO] AnonymousA irworthiness requirements

[Jategaonkar'93] Jategaonkar, R.V.Identification of actuation systems and aerodynamic effects of direct-lift-controlflapsIn Journal of Aircraft, Vo1.30, No.5 DLR, Braunschweig, 1993, Institute of FlightMechanics

[Kruijsen'94] Kruijsen, E.A.C.Lr2pr flight test manualInterim report, Delft, 1994, Delft University of Technology

[Kubbat] Kubbat, W.J.Investigations on direct force control for CCV aircraft during approach and landingin AGARD take-off and landingMesserschmitt-Boelkow-Blohm G.m.b.H., Munisch

[Van der Linden'94] Van der Linden, C.Implementation of the aerodynamics of the Cessna Citation in Simulink(Computer program) Delft, 1994, Delft University of technology

[McLean'90] McLean, D.A utomatic flight control systemsPrentice Hall International (UK), London, 1990

[Ruigrok'92] Ruigrok, R.C.J.Ontwerp en evaluatie van twee automatische landingssystemen voor de Cessna Ci­tation 500, gebruik makend van ILS en MLSMaster's Thesis, Delft, 1992, Delft University of Technology

[Sachs'83] Sachs, G.Direct force controlin AGARD course on aerodynamic characteristics of controlHochschule der Bundeswehr, Munich, 1983

[Schanzer] Schanzer, G.Direct lift control for flight path control and gust alleviationin AGARD guidance and control design considerations for low-altitude andterminal-area flightUniversity of Technology Brunswick

- 51 -~ Bibliography

@J----------,Jt:::.

[Simulink] The Mathworks inc.Simulink dynamic system simulation software, user's guide

[Sobel'84] Sobel, K.M. et al.Synthesis of direct lift control laws via eigenstrueture assignmentIEEE, New York, 1984, Lockheed-California Company

[Thomas'83] Thomas, H.H.B.M. (consultant)The aerodynamics of aircraft control, a general survey in the context of active con­trol technologyin AGARD course on aerodynamic characteristics of control, report nr.711, july1983

[Van der Vaart'93] Vaart, J.C. van der, and Mulder, J.A.Aircraft responses to atmospheric turbulenceLecture notes D-47, Delft, 1993, Delft University of Technology

[Wilkin'82] Wilkin, D. and Stiles, H.Optimizing the F-14 DLC/APC system for improved glideslope performanceGrumman Aerospace Corporation Calverton, New York, stability and control sec­tion, 1982

- 53 -A Linearized aircraft model~ --------------------------------~

Appendix A

Linearized aircraft model

T~ble A.1: (Confidential) State matrix

The state equation of the twelve state model is given by ;f = A/in±.+B/in'lJ,

with state vector ±. = [V 0: () q h x (3 cP p r 'ljJ y]T and input vector'lJ = lUg O:g be b1 bsb TXj TX2 ba brf. The state vector is reordered in a partrepresenting the symmetric motion and a part representing the asymmetricmotion.

Table A.2: (Confidential) Input matrix

The input vector is reordered in a part representing symmetric controls anda part representing asymmetric controls.

Table A.3: (Confidential) Output matrices

From G/in the non-identity part is shown, from D/in the non-zero part. Theoutput equation of the twelve state model is given by: l!.. = Glin±. + D/in'lJ

wi th l!.. = [±. h h q] T

- 55 -Data plots of stability derivatives

~s------------.J'

Appendix B

Data plots of stability derivatives

Figure B.1: (Confidential) cdsaf: Increment in drag coefficient due to full bsb

Figure B.2: (Confidential) clsam: Increment in lift due to full bsb

Figure B.3: (Confidential) cmesaf: Increment in Cm6 due to full be.b

- 56 -Direct lift control for the Cessna Citation II,./e ----------------------

,,)'

Figure B.4: (Confidential) cmqsaf: Increment in Cm6 due to pitch ratesb

Figure B.5: (Confidential) cmsam: Incr. in pitching moment due to full /)sb

Figure B.6: (Confidential) kss: Speedbrake effectiveness factor

- 57 -~ Simulink modelse---------­~

Appendix C

Simulink models

e.l Description of modifications

Some modifications to the Simulink model are made so that the speedbrake and flapdeflection can be given as an input of the model. The model is built up in layers, eachrepresenting a small subsystem of the aircraft's aerodynamics. The uppermost layeracts as an input/output layer for control signals and measured values. From this layer,one can get to the dynamics of the Cessna Citation, which are calculated in layer twoand lower. These layers come back in all models, i.e. the trim model (section 3.2), thesimulation model (section 3.4) and the linearization model (section 3.3). The uppermostlayer of these models will be discussed in the next sections. The modifications in thecommon underlying layers will be discussed here. The modified parts of the models areon computer simulation drawn in green and on paper drawn with dotted lines.

A complete overview of all layers in the Simulink model is given in Figure C.l. The layernames printed in a grey box have been modified. The subsystem CITATION (Figure C.5)is not really modified, but it feeds the enlarged input vector through to the subsystemAFM. The input vector is enlarged from eight to nine inputs by adding the speedbrakedeflection command to it. Subsystem AFM (Figure C.6) feeds the input vector through tothe rudder to rudder-trim surface coupling block (rud-rudtrim coupling). Within thisblock (Figure C.7) the input vector is decomposed into separate lines. Both multiplexerand demultiplexer are enlarged by one control input. Because the speedbrakes do notinteract with the rudder, the speedbrake control input is directly connected to themultiplexer and the input vector is composed again and fed to the aeromod subsystem(Figure C.8). Within this block, the input vector is decomposed and fed to subsystemsfor calculating the increments in drag, lift and moment coefficients. The input signalfor the speedbrakes is kept apart for clearness. The vector, created at the beginning ofthe block, is given by:

- 58 -~ Direct lift control for the Cessna Citation II

<§J------------~

where A~cq = 0.3 - xCgr~f-6. Within this layer, the aerodynamic forces and momentsare calculated. For each direction, there is a separate block. The blocks in whichthe speedbrakes take effect are CDCAL, CLCAL and CMCAL. In CDCAL (Figure C.g), theincrement in drag due to a speedbrake deflection is calculated, according to (3.4). InCLCAL (Figure C.lO), the increment in lift due to a speedbrake deflection is calculated,according to (3.5). Because the calculation of the increment in pitching moment dueto a speedbrake deflection Figure C.l! is somewhat more complicated, a subsystemSpeed Brakes (Figure C.l2) is added to CMCAL. From the speedbrake deflection angle,the angle of attack, the elevator deflection angle and the pitch rate, the increment inpitching moment due to a speedbrake deflection is calculated. The tables, used in CDCAL,CLCAL and CMCAL are loaded into the Matlab workspace by typing:load data\citdlc.rnat

Figure C.l: (Confidential) Overview of Simulink model layers

The layers displayed on a grey background are modified. Layer 1 is thehighest level. Layer 2 is the basis of the main aerodynamics model of theCessna Citation.

Figure C.2: (Confidential) cit-trim: citation trim model

This model is part of the Citation trim routine. The output of this routineis a trimmed operation point (xo, uo) that can be used with linearization andsimulation studies. The input vector of this model is 'Jl. = [Oe Oa Or TXl Tx2 ]T.The outputs of this model are the states of the state space model. The block'Citation' is the second layer Figure C.5.

Figure C.3: (Confidential) cit-lin: citation linearization model

This model is part of the Citation linearization routines. The input of themodel is a trimmed flight condition, generated by ciLtrirn. The outputsare the output vector of the state space model.

, - 59 -~ Simulink modelsEJ------------,jt::.

Figure C.4: (Confidential) cit-ctrl: citation simulation model

This model is used for simulation studies. The output equation (sensors),the output filter (filters), the controllers and actuator dynamics form aclosed loop with the Citation aerodynamics. The turbulence generator isdesigned in section 4.2.

Figure C.5: (Confidential) CITATION: Main aerodynamics model

This is the uppermost layer of the aerodynamics model of the Cessna Cita­tion. From within this block the entire aircraft state is calculated.

Figure C.6: (Confidential) AFM: Calculation of the aerodynamics coefficients

From within this block all aerodynamics coefficients are calculated(aeromod)

Figure C.7: (Confidential) rud-rudtrim coupling

A coupling between the rudder deflection angle and the rudder trim deflec­tion angle.

Figure C.S: (Confidential) aeromod: aerodynamics model of gust and controls

From within this block the aerodynamics in all directions are calculated. Inthe blocks CDCAL, CLCAL and CMCAL the drag, lift and moment coefficientsbased on control surface deflection are calculated.

Figure C.9: (Confidential) cdcal: calculation of the drag coefficient

Calculation of the drag coefficient as a function of the control surface deflec­tions, according to (3.1) and (3.4).

Figure C.10: (Confidential) clcal: calculation of the lift coefficient

Calculation of the lift coefficient as a function of the control surface deflec­tions, according to (3.1) and (3.5).

- 60 -~ Direct lift control for the Cessna Citation IIe----------~

Figure C.lI: (Confidential) cmcal: calculation of the moment coefficient

Calculation of the pitching moment coefficient as a function of the controlsurface deflections, according to (3.1) and (3.6).

Figure C.12: (Confidential) Speedbrakes contribution to pitching moment

Calculation of the contribution of the speedbrakes to the pitching momentcoefficient, according to (3.6)

Figure C.13: (Confidential) Turbulence generator

Calculation of the horizontal gust velocity and gust induced angle of attackusing the state space model (4.29) with white noise sources. The outputsare added to the gust vector.

Figure C.14: (Confidential) actuator dynamics

This model represents the actuator dynamics and the deflection rate anddeflection angle limiters. Overall gains are placed before the actuators toprevent saturation.

Figure C.15: (Confidential) Linear model of DLC configured aircraft

This is the linear simulation model. With this model the time dependent out­puts resulting from gust input can be analysed. In turbulence the horizon­tal gust velocity and gust induced angle of attack are calculated. controlscontains 'scopes' for plotting the input variables. In Calculate place theplace (x-direction) of the aircraft is calculated. In filters the outputs arebandpass filtered.

Figure C.16: (Confidential) Elevator controller

This model represents the elevator controller, with proportional feedback ofthe pitch angle (0), the pitch rate (q) and the pitch acceleration (q). Anextra input, called !:l0 is added for correction of the moment induced by thedirect lift control surfaces.

- 61 -~ Simulink models<@y-----------------,Jt:::.

Figure C.17: (Confidential) Flaps controller

This model represents the flaps controller as well as the speedbrakes con­troller, which have the same structure. The vertical speed (il,), vertical ac­celeration (h) and the pitch acceleration (q) are fed back to the flaps andthe elevators.

- 63 -Open loop system characteristicse----------,)'

Appendix D

Open loop system characteristics

The labels of the curves in the plots of each figure are described in the subtitle of eachfigure. The labels are given per plot, seen in top to bottom order. So,' h, h, q , meansthat the uppermost curve represents h, the second curve h and the third curve q.

Figure D.l: (Confidential) System poles, short period mode

Figure D.2: (Confidential) System poles, phugoid mode

Table D.l: (Confidential) Open loop system poles

- 64 -Direct lift control for the Cessna Citation II/'e ----------------------

~

Figure D.3: (Confidential) Actuator with washout filter, step response

Top to bottom, resp: no washout (Tw =(0), Tw =10, Tw =8, Tw =6, Tw =4, Tw =2 sec.

Figure D.4: (Confidential) Dryden spectra: [WI, W3] to lUg, a g]

Top to bottom, resp: U g and a g

Figure D.5: (Confidential) Bode plot U g to h, hand q

Top to bottom, resp: h, h, qand h, h, q

Figure D.6: (Confidential) Bode plot a g to h, hand q• u ...

Top to bottom, resp: h, h, qand h, q, h

Figure D.7: (Confidential) Bode plot De to V and ()

Top to bottom, resp: V, () and V, ()

Figure D.8: (Confidential) Bode plot De to a and q

Top to bottom, resp: q, a and a, q

Figure D.9: (Confidential) Bode plot De to h, hand q

. .. ...Top to bottom, resp: h, h, qand h, q, h

Figure D.10: (Confidential) Bode plot DJ to V and ()

Top to bottom, resp: V, () and (), V

- 65 -~ Open loop system characteristics

<@9----------~

Figure D.ll: (Confidential) Bode plot of to a and q

Top to bottom, resp: q, a and a, q

Figure D.12: (Confidential) Bode plot of to h, hand q

.. .. .. ..Top to bottom, resp: h, h, q and h, h, q

Figure D.13: (Confidential) Bode plot Osb to V and ()

Top to bottom, resp: V, () and V, ()

Figure D.14: (Confidential) Bode plot Osb to a and q

Top to bottom, resp: q, a and q, a

Figure D.15: (Confidential) Bode plot Osb to h, hand q

Top to bottom, resp: h, h, q and h, q, h

- 67 -..<. Closed loop system characteristics

@----------~

Appendix E

Closed loop system characteristics

- 68 -Direct lift control for the Cessna Citation II

( 0.5 0.6 0.7 0.8 0.9 ~1

J{ee -0.04 -0.06 -0.08 -0.10 -0.11 -0.13

Table E.1: Pitch angle to elevator feedback gains

8 .u;+k'y: ' de' to 'theta' wn step 2.4

-2 ......, ; ," .,": .:::

-4 .,/_. ...••.•.••..........~/_ .•... /.'

.......:,: .' .:" ,:-6 . ....:: ....:< .../.. :-8

1..... ... " ,

~ ~ 4 ~ aReal Axis

2 4 6

Figure E.1: Root locus elevator to pitch angle

u;+k'y: ' de' to 'theta' wn step 2.40.15r:---,.....,,-----,..,...--,----,---...------,r------,----,

0.1

0.05

-0.05

-0.1

0.05 0.1 0.15

Figure E.2: Root locus elevator to pitch angle, phugoid mode

- 69 -A Closed loop system characteristics~ -----------------------------------~

( 0.5 0.6 0.7 0.8 0.9 ~1

]{eQ -0.03 -0.07 -0.11 -0.15 -0.21 -0.25

Table E.2: Pitch rate to elevator feedback gains

u=+k'y: ' de' 10 'q 'wn Slep 2.46r:--...".,.,..---..,.-,.........,---..,.---..,.-,--.....---...,....-------,

4

2

~~ oH-----'---+--...;-----E) ..... -:-;......

.E

-2

-~6'-'---"-'-'--4-----'--~--'-'-----'------'-----'------'2-----'-4----J6·

Figure E.3: Root locus elevator to pitch rate

u=+k'y: , de' 10 'q 'wn slep 2.40.15r:--......-,---..,.-,.........,---..,.---,---.....---...,....------,

0.1

0.05

.!!l

~ 0~······ ·.. ·.. ·.. ·.. · ···· · <3· .. ·::[fi>· .. · .. · .. · ·· ~'".E

-0.05

-0.1

-0·!6.~15-----'-------'----:!0.:-1 -----'--'--:::-'::--'------'--:--------:-'0.0=-=-5--::'-0.1;---70..15

Figure E.4: Root locus elevator to pitch rate, phugoid mode

- 70 -Direct lift control for the Cessna Citation IIw"e ----------------------

~

poles ( Wn

-0.0750±0.0563j 0.7998 0.0937-3.9408±3.5321j 0.7447 5.2920

Table E.3: System poles when inner loop closed

Short period modes, inner loop active4

x

3

2

"'xg,oE

-1

-2

-3

x

-4-4 -3.5 -3 -2.5 -2 -1.5 -1

Real Axis

Figure E.5: System poles, short period mode, inner loop active

phugoid modes, inner loop active

0.15

0.1

0.05

O'~E

-0.05

-0.1

-0.15

-0.2.CLC._L_--,-J:-~'-'---_.J,..,----'-'- _ __'__ _'______L____'__...L.:l-0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -o.Q1 0 0.Q1

Real Axis

Figure E.6: System poles, phugoid mode, inner loop active

- 71 -A. Closed loop system characteristics~ -----------------------------------~

( 0.80 0.85 0.90 0.95 ~1

Kf;, 0.11 0.19 0.24 0.34 0.37

Kf;, 0.13 0.21 0.36 0.38

Table E.4: Vertical speed and acceleration to flaps feedback gains

u=-k'y: ' dr 10 'hdOI' wn step 2.48,..."--.",.,,...---.,.~---.,......_c__:_--r-----r'-___.--.______,

4

-2

-4

Figure E. 7: Root locus flaps to vertical speed

u=-k'y: ' dr 10 'az ' wn slep 1.40.15.----....,.....,.-..,..----r----r----.-----r-----,

0.150.10.05

-0.1

0.1

0.05

-0.05

~~ of-+--+-+-------e'*------------i.5

Figure E.8: Root locus flaps to vertical acceleration

- 72 -Direct lift control for the Cessna Citation II~e --------------------------------

~

( 0.80 0.85 0.90 0.95 ~1

KIa 0.12 0.22 0.30 0.43 0.45

K sb ;. -0,02 -0.03 -0.06 -0.07

Table E.5: Flaps and speedbrake feedback gains

u--k'y: ' df' to 'qdol' wn step 1.4

4

-4

-6

-6 -4 o 2Real Axis

4 6

Figure E.9: Root locus flaps to pitch acceleration

- 73 -Closed loop system characteristics

( 0.80 0.85 0.90 0.95 ~1

K sb - -0.93 -1.59 -2.06 -2.52 -2.66K sba 1.12 1.83 2.38 2.87 3.08

Table E.6: Pitch rate to elevator feedback gains

u=+k"y: 'dsp' to 'az ' wn step 2.8

8642·8

2

4

-4

~'" of--,---+-+--'--+-.:,......~----------1'".§

-2

Figure E.I0: Root locus speedbrakes to vertical acceleration

u=-k"y: 'dsp' to 'qdot' wn step 1.4

6

4

2

-4

·6

"'. ". . , .

. ". ".'- : .

....•:.. .....:--:'" .: ." ":-..,.\. '.:"... ~:

-8

~ ~ 4 ~ 0 2Real Axis

4 6 8

Figure E.1l: Root locus speedbrakes to pitch acceleration

- 74 -Direct lift control for the Cessna Citation II

N·,.~,--:'" o~·····.·~ :;·;.:;.;'--=i==!::::7'~Z~~~>.:-;:i'---···:::··:···:':·i~u ~I::

~ -20~""""':' :;.;.:;-:-; : ••. ::.;.:;; ......•.. ; .. ; ..•• :""'.... :,,,,; : :;;

ClQl

~ ·180

l'!c.

·360

.. .... . ..

~: ::::::::::

.~~~...•...~.. q'~

10°Frequency (rad/sec)

10'

Figure E.12: Bode plot 8e to () and q

Top to bottom, resp: (), q and q, ()

flaps 10 (hdol,hddol,qdol)50r-~~~~...--~~~~.....--~~......,---~.......,

'" 0UI::'OJC!l -50

· .. [ill[ . .... .... .· . . . . . . . . . . . . .· . . .. . .. .... .· . . . . . . . . . . . .· . . . . . . . . . . . . .· . . . . . . . . . . . . . .qqq,,;: =n=n+=-... ~,..:....~ ... ,

-100 '-:---'----'---'--'--'-'-'--'-'-:----'---'---'--'-'-'--'-'-'-:,----'---'----'--'-'-'-'-i.i-,---'--'------'--'-'-'-'J10" 10" 10° 10' 10'

Frequency (rad/sec)

720

ClQlU

mO·· ."qq. ':"3;--;~...~q.q.~.......==.~~jg :~C. ~

. . . . . ... .·720

10°Frequency (rad/sec)

10' 10'

Figure E.13: Bode plot 81 to h, hand q

Top to bottom, resp: h, h, qand h, h, q

- 75 -Closed loop system characteristics

speedbrakes to (hdot,hddot,qdot)50

'"0

"0c:'iiiCl

·50

-10010" 10°

Frequency (rad/sec)10'

360 ' " .

o .

-360

10°Frequency (rad/sec)

10'

Figure E.14: Bode plot Dsb to hand h

• u • ..

Top to bottom, resp: h, h, qand h, h, q

Impulse: ' de' to 'theta'1

I f\ ~

I V \../ ----1 III

o

0.5

-0.5..ena .fil­~'1.5"0

"""~ ·2

'"-2.5

·3

-3.5

-4o 50 100 150 200 250 300 350 400 450 500time

Figure E.15: Impulse response elevator to pitch angle

Open loop and closed loop

- 76 -Direct lift control for the Cessna Citation II

Impulse: ' de' to 'q ,10

5

o

1\

iN"--/'

-20

-25

-30o 0.5 1.5 2 2.5 3time

3.5 4 4.5 5

Figure E.16: Impulse response elevator to pitch rate

Open loop and closed loop

Impulse: ' df' to 'hdot '

\\

\"

............

20

25

o

5

0

50 10 20 30 40 50 60 70 80 90 100

time

15~§

~.. 10

~1i.~

Figure E.17: Impulse response flaps to vertical speed

Open loop and closed loop

- 77 -~ Closed loop system characteristics

<@y----------~

Impulse: ' df to 'az '18.----,~---,----.-___r-____.----r--___._-__,__-_r___

16H1\ , : : ; ; ; : : ,

14

12

'"'"§ 10c.:«;;; 8r ..\.... ·:.... ·.. ·.. :·· .. ·......;· .... ·....:· .... ·· .. ,· .... ·.. ·:· .... ·-g

i~§E :: -: :- ;. .·2~----='':----'-----=-'=-----7--='::------:----=''=---7---:'=-------:o 0.5 1.5 2 2.5 3 3.5 4 4.5 5

lime

Figure E.IS: Impulse response flaps to vertical acceleration

Open loop and closed loop

Impulse: 'dsp' to 'hdot '

r1 \

{. /\ ~

\/ 'J '--'" .

1 v

3 IIV

4

o3li5c.~"'­'0~C.E..

·2

2

o 50 100 150 200 250 300 350 400 450 500time

Figure E.19: Impulse response speedbrakes to vertical speed

Open loop and closed loop

- 78 -Direct lift control for the Cessna Citation II

Impulse: 'dsp' to 'qdot .1.2

0.8

Q)8 0.6Q.<I>

~ 0.4'0

..EQ.~ 0.2

o

~

1/\\\

}

~~.... ·· ... V...

•

0.5 1.5 2 2.5 3 3.5time

4 4.5 5

Figure E.20: Impulse response speedbrakes to pitch acceleration

Open loop and closed loop

- 79 -~ Results of linear simulations

@9------------~

Appendix F

Results of linear simulations

- 80 -Direct lift control for the Cessna Citation II

speed angle of attack

50 100time

..~

£!...

-200 50 100 150 200

time

p~ch angle p~ch rate0.1 0.04

0.02

~ .,. 005

Figure F.l: Response of V, (Y, (), q to gust (inner loop active)

vertical speed vertical accelieration15 0.5

15"0.c

-10 -0.500 100 150 50 100 150 200

time time

flight path angle Vert.Acc at pilot seat0.1

..E <E !;;!2l.

-0.10 50 100 150 200 50 100 150 200

time time

Figure F.2: Response of h, h", q to gust (inner loop active)

- 81 -Results of linear simulations

elevator tlaps0.04

0.5

~I -,0 ~ 0., .,

" "-{l.5

-0.04 -10 50 100 150 200 0 50 100 150 200

time time

speedbrakes1

0.5c..,~I 0-8

-0.5

10 50 100 150 200

time

Figure F.3: Control activity (inner loop active)

speed30,-----------,

>

p~ch angle0.02,-----------,

0.01

ClI

10 0:;

50 100 150 200time

angle ot attack

pitch rate

Figure FA: Response of V, 0', 0, q to gust (DLC loop active)

- 82 -Direct lift control for the Cessna Citation II

vertical speed vertical accelleration2

{j 0"'"

-1

-20 50 100 150 200 50

time

flight path angle Vert.Acc at pilot seat

ClIEEClI 00>

Figure F.5: Response of h, h", q to gust (DLC loop active)

elevator flaps

0.05

~I~ 0

o 50 100 150 200time

speedbrakes

0.2.------------,

0.1 .

-, 0ClI

~-0.1 .

-0.21-1·· ·, III· ..> : ,

-o.30:--~SO::---1,.:.0:-0--'-1-:CSO::---:-'.2oo

time

o.2r-----------,

c.VI

[4' 0~

-0.20'-----'50--1.....0-0--1.....50--200

time

Figure F .6: Control activity (DLC loop active)

- 83 -..<. Results of linear simulations<§----------~

Condition Vt a () q h h q I aZA

[s~~\ ] [rad] [rad] [Tad] [sr:!:1 ] Lr:!:2] [Tad] [rad] [sr:!:2]s-1 s-2h=5000, M=O.4 2.938 0.008 0.021 0.010 3.460 0.101 0.044 0.027 0.100h=5000, M=0.5 5.372 0.008 0.024 0.012 4.950 0.152 0.064 0.031 0.151h=5000, M=0.6 7.807 0.009 0.029 0.014 7.082 0.215 0.092 0.037 0.216

Table F.1: Reference RMS values

h=5000m relative valuesMach 0.4 Vt a () q h h q I aZA J RDNo controller 987 170 1445 357 1095 443 106 1095 446 9.7Inner loop 100 100 100 100 100 100 100 100 100 3.7Elevator loop 179 119 26 159 23 82 171 23 60 3.4Flap+elevator 210 58 23 115 16 55 136 16 55 2.9Speedbrake+elev. 174 117 26 163 22 83 175 22 60 3.5DLC+elevator 206 58 23 117 15 56 138 15 55 2.9

h=5000m relative valuesMach 0.4 Vt a () q h h q I aZA JRDNo controller 870 151 1788 363 1393 442 104 1393 442 13.7Inner loop 100 100 100 100 100 100 100 100 100 4.6Elevator loop 152 115 22 175 19 83 185 19 57 4.3Flap+elevator 176 44 19 110 12 50 133 12 52 3.3Speedbrake+elev. 147 114 22 180 19 84 190 19 57 4.3DLC+elevator 172 44 19 112 12 50 134 12 52 3.3

h=5000m relative valuesMach 0.4 Vt a () q h h q I aZA JRDNo controller 754 137 1770 347 1377 428 102 1377 425 18.1Inner loop 100 100 100 100 100 100 100 100 100 5.8Elevator loop 144 113 22 171 16 75 177 16 54 5.0Flap+elevator 161 37 19 103 9 41 125 9 50 3.6Speedbrake+elev. 138 112 22 177 16 77 182 16 54 5.1DLC+elevator 157 36 19 105 9 42 127 9 50 3.7

Table F.2: Relative results of linear simulations

- 84 -..<. Direct lift control for the Cessna Citation IIe----------~

h=5000m RMS values peak valuesMach 0.5 8e 8f 8sb 8e 8f 8sb

No controller 0.000 0.000 0.000 0.000 0.000 0.000Inner loop 0.012 0.000 0.000 0.033 0.000 0.000Elevator loop 0.014 0.000 0.000 0.045 0.000 0.000Flaps+elevator 0.012 0.071 0.000 0.036 0.215 0.000Speedbrake+elev. 0.014 0.000 0.066 0.044 0.000 0.219DLC+elevator 0.011 0.070 0.046 0.036 0.213 0.152

h=5000m RMS values peak valuesMach 0.5 8e 8f 8sb 8e 8f 8sb

No controller 0.000 0.000 0.000 0.000 0.000 0.000Inner loop 0.014 0.000 0.000 0.039 0.000 0.000Elevator loop 0.016 0.000 0.000 0.051 0.000 0.000Flaps+elevator 0.013 0.080 0.000 0.041 0.248 0.000Speedbrake+elev. 0.016 0.000 0.078 0.050 0.000 0.257DLC+elevator 0.013 0.080 0.053 0.041 0.247 0.169

h=5000m RMS values peak valuesMach 0.6 8e 8f 8sb 8e 8f 8sb

No controller 0.000 0.000 0.000 0.000 0.000 0.000Inner loop 0.017 0.000 0.000 0.046 0.000 0.000Elevator loop 0.019 0.000 0.000 0.061 0.000 0.000Flaps+elevator 0.015 0.088 0.000 0.047 0.262 0.000Speedbrake+elev. 0.019 0.000 0.096 0.061 0.000 0.288DLC+elevator 0.015 0.087 0.059 0.047 0.262 0.199

Table F .3: Control surface deflections

- 85 -A. Results of linear simulationsce;----------------------------,)I:::.

Condition Vt a () q h h q I aZA

[sr:!.l ] [rad] [rad] [Tad] [sr:!.l ] [sr:!.2 ] [Tad] [rad] [sr:!.2 ]s-l s-2h=10km, M=O.4 0.973 0.008 0.016 0.008 2.593 0.059 0.026 0.022 0.058h=lOkm, M=0.6 3.657 0.007 0.020 0.011 4.566 0.125 0.049 0.025 0.122h=1000, M=0.3 2.198 0.008 0.021 0.010 2.779 0.084 0.038 0.028 0.085

Table F.4: Reference RMS values

h=10km relative valuesMach 0.4 Vt a () q h h q I aZA J RDNo controller 1281 172 894 262 650 382 102 650 389 5.8Inner loop 100 100 100 100 100 100 100 100 100 2.9Elevator loop 272 136 43 200 26 89 204 26 59 2.8Flaps+elevator 305 80 34 169 17 70 181 17 56 2.6Speedbrake+elev. 267 134 43 202 25 90 207 25 59 2.9DLC+elevator 302 80 34 171 17 70 183 17 57 2.6

h=10km relative valuesMach 0.4 Vt a () q h h q I aZA J RDNo controller 1097 173 1687 314 1307 442 108 1307 449 11.6Inner loop 100 100 100 100 100 100 100 100 100 4.1Elevator loop 166 137 40 307 18 120 311 18 58 4.8Flaps+elevator 186 56 23 158 12 63 175 12 53 3.4Speedbrake+elev. 161 140 42 327 18 127 331 18 59 5.0DLC+elevator 182 56 24 162 12 65 179 12 54 3.4

h=lOkm relative valuesMach 0.4 Vt a () q h h q I aZA JRDNo controller 978 150 1308 409 989 419 105 989 416 8.1Inner loop 100 100 100 100 100 100 100 100 100 3.4Elevator loop 199 122 22 126 27 78 143 27 65 3.1Flaps+elevator 243 63 23 101 19 57 123 19 60 2.8Speedbrake+elev. 194 120 22 128 27 79 144 27 65 3.1DLC+elevator 239 61 23 101 19 57 124 19 60 2.8

Table F .5: Relative results of linear simulations

- 86 -..<.. Direct lift control for the Cessna Citation IIe----------~

h=lOkm RMS values peak valuesMach 0.4 8e 81 8sb 8e 81 8sb

No controller 0.000 0.000 0.000 0.000 0.000 0.000Inner loop 0.009 0.000 0.000 0.025 0.000 0.000Elevator loop 0.012 0.000 0.000 0.037 0.000 0.000Flaps+elevator 0.009 0.057 0.000 0.028 0.163 0.000Speedbrake+elev. 0.011 0.000 0.055 0.036 0.000 0.173DLC+ elevator 0.009 0.057 0.037 0.028 0.162 0.118

h=10km RMS values peak valuesMach 0.6 8e 81 8sb 8e 81 8sb

No controller 0.000 0.000 0.000 0.000 0.000 0.000Inner loop 0.012 0.000 0.000 0.031 0.000 0.000Elevator loop 0.014 0.000 0.000 0.043 0.000 0.000Flaps+elevator 0.012 0.072 0.000 0.036 0.219 0.000Speedbrake+elev. 0.014 0.000 0.071 0.043 0.000 0.199DLC+elevator 0.011 0.071 0.048 0.036 0.217 0.159

h=1000m RMS values peak valuesMach 0.3 8e 81 8sb 8e 81 8sb

No controller 0.000 0.000 0.000 0.000 0.000 0.000Inner loop 0.012 0.000 0.000 0.034 0.000 0.000Elevator loop 0.014 0.000 0.000 0.044 0.000 0.000Flaps+elevator 0.012 0.071 0.000 0.037 0.208 0.000Speedbrake+elev. 0.014 0.000 0.065 0.044 0.000 0.208DLC+elevator 0.012 0.070 0.047 0.037 0.207 0.152

Table F.6: Control surface deflections

- 87 -~ Matlab filese----------

..J'

Appendix G

Matlab files

Confidential